superlinear parabolic problems: blow-up, global existence and …kaplicky/pages/pages/... · 2013....

Birkhäuser Advanced Texts

Edited byHerbert Amann, Zürich UniversitySteven G. Krantz, Washington University, St. LouisShrawan Kumar, University of North Carolina at Chapel Hill

Superlinear Parabolic ProblemsBlow-up, Global Existence and Steady States

BirkhäuserBasel · Boston · Berlin

Authors:

Slovakia France

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Basel · Boston · Berlin

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. MODEL ELLIPTIC PROBLEMS

2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. Classical and weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5. Pohozaev’s identity and nonexistence results . . . . . . . . . . . . . . . . . . . . . . . . . . . 186. Homogeneous nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207. Minimax methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298. Liouville-type results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369. Positive radial solutions of ∆u + up = 0 in R

n . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10. A priori bounds via the method of Hardy-Sobolev inequalities . . . . . . . . . . 5511. A priori bounds via bootstrap in Lp

δ-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112. A priori bounds via the rescaling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513. A priori bounds via moving planes and Pohozaev’s identity . . . . . . . . . . . . . 68

II. MODEL PARABOLIC PROBLEMS

14. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515. Well-posedness in Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516. Maximal existence time. Uniform bounds from Lq-estimates . . . . . . . . . . . . 8717. Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118. Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

19. Global existence for the Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121. Small data global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2. Structure of global solutions in bounded domains . . . . . . . . . . . . . . . . . . . 120

3. Diffusion eliminating blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

20. Global existence for the Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291. Small data global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2. Global solutions with exponential spatial decay . . . . . . . . . . . . . . . . . . . . . 137

3. Asymptotic profiles for small data solutions . . . . . . . . . . . . . . . . . . . . . . . . . 139

21. Parabolic Liouville-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15022. A priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

1. A priori bounds in the subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

2. Boundedness of global solutions in the supercritical case . . . . . . . . . . . . . 166

3. Global unbounded solutions in the critical case . . . . . . . . . . . . . . . . . . . . . . 171

4. Estimates for nonglobal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

vi

23. Blow-up rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17724. Blow-up set and space profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

25. Self-similar blow-up behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19526. Universal bounds and initial blow-up rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20227. Complete blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21828. Applications of a priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

1. A nonuniqueness result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

2. Existence of periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

3. Existence of optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4. Transition from global existence to blow-up and stationary solutions . 237

5. Decay of the threshold solution of the Cauchy problem . . . . . . . . . . . . . . 239

29. Decay and grow-up of threshold solutions in the super-supercritical case 245

III. SYSTEMS

30. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25131. Elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

1. A priori bounds by the method of moving planes and Pohozaev-type

identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

2. Liouville-type results for the Lane-Emden system . . . . . . . . . . . . . . . . . . . 260

3. A priori bounds by the rescaling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

4. A priori bounds by the Lpδ alternate bootstrap method . . . . . . . . . . . . . . 266

32. Parabolic systems coupled by power source terms . . . . . . . . . . . . . . . . . . . . . . 2721. Well-posedness and continuation in Lebesgue spaces . . . . . . . . . . . . . . . . . 273

2. Blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

3. Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

4. Blow-up asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

33. The role of diffusion in blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2871. Diffusion preserving global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

2. Diffusion inducing blow-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

3. Diffusion eliminating blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

IV. EQUATIONS WITH GRADIENT TERMS

34. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31335. Well-posedness and gradient bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31436. Perturbations of the model problem: blow-up and global existence . . . . . . 31937. Fujita-type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

38. A priori bounds and blow-up rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33839. Blow-up sets and profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

vii

40. Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary 3551. Gradient blow-up and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

2. Asymptotic behavior of global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

3. Space profile of gradient blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

4. Time rate of gradient blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

41. An example of interior gradient blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

V. NONLOCAL PROBLEMS

42. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37743. Problems involving space integrals (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377


2. Blow-up rates, sets and profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

3. Uniform bounds from Lq-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

4. Universal bounds for global solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

44. Problems involving space integrals (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3981. Transition from single-point to global blow-up . . . . . . . . . . . . . . . . . . . . . . . 398

2. A problem with control of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

3. A problem with variational structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

4. A problem arising in the modeling of Ohmic heating . . . . . . . . . . . . . . . . 412

45. Fujita-type results for problems involving space integrals . . . . . . . . . . . . . . . 41846. A problem with memory term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421


2. Blow-up rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

APPENDICES

47. Appendix A: Linear elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4291. Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

2. Lp-Lq-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

3. An elliptic operator in a weighted Lebesgue space . . . . . . . . . . . . . . . . . . . 434

48. Appendix B: Linear parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4381. Parabolic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

2. Heat semigroup, Lp-Lq-estimates, decay, gradient estimates . . . . . . . . . 439

3. Weak and integral solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

49. Appendix C: Linear theory in Lpδ-spaces and in uniformly local spaces . . 447

1. The Laplace equation in Lpδ-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

2. The heat semigroup in Lpδ-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

3. Some pointwise boundary estimates for the heat equation . . . . . . . . . . . 452

4. Proof of Theorems 49.2, 49.3 and 49.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

5. The heat equation in uniformly local Lebesgue spaces . . . . . . . . . . . . . . . 460

viii

50. Appendix D: Poincare, Hardy-Sobolev, and other useful inequalities . . . . 4621. Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

2. The Poincare inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

3. Hardy and Hardy-Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

51. Appendix E: Local existence, regularity and stability for semilinear para-

bolic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4661. Analytic semigroups and interpolation spaces . . . . . . . . . . . . . . . . . . . . . . . 466

2. Local existence and regularity for regular data . . . . . . . . . . . . . . . . . . . . . . 470

3. Stability of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

4. Self-adjoint generators with compact resolvent . . . . . . . . . . . . . . . . . . . . . . 488

5. Singular initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

6. Uniform bounds from Lq-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

52. Appendix F: Maximum and comparison principles. Zero number . . . . . . . 5071. Maximum principles for the Laplace equation . . . . . . . . . . . . . . . . . . . . . . . 507

2. Comparison principles for classical and strong solutions . . . . . . . . . . . . . 509

3. Comparison principles via the Stampacchia method . . . . . . . . . . . . . . . . . 512

4. Comparison principles via duality arguments . . . . . . . . . . . . . . . . . . . . . . . . 515

5. Monotonicity of radial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

6. Monotonicity of solutions in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

7. Systems and nonlocal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

8. Zero number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

53. Appendix G: Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52854. Appendix H: Methodological notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Introduction

This book is devoted to the qualitative study of solutions of superlinear elliptic andparabolic partial differential equations and systems. Here “superlinear” means thatthe problems involve nondissipative terms which grow faster than linearly for largevalues of the solutions. This class of problems contains, in particular, a number ofreaction-diffusion systems which arise in various mathematical models, especiallyin chemistry, physics and biology.

For parabolic problems of this type it is known that a solution may cease toexist in a finite time as a consequence of its L∞-norm becoming unbounded: Thesolution blows up. On the other hand, in many of these problems there exist alsoglobal solutions (in particular, stationary solutions). Both global and blowing-up solutions may be very unstable and they may exhibit a rather complicatedasymptotic behavior.

Concerning elliptic problems, we consider questions of existence and nonexis-tence, multiplicity, regularity, singularities and a priori estimates. Special emphasisis put on those results which are useful in the investigation of the correspondingparabolic problems. As for parabolic problems, we study the questions of local andglobal existence, a priori estimates and universal bounds, blow-up, asymptotic be-havior of global and nonglobal solutions.

The study of superlinear parabolic and elliptic equations and systems has at-tracted the attention of many mathematicians during the past decades. Althougha lot of challenging problems have already been solved, there are still many openquestions even in the case of the simplest possible model problems. Unfortu-nately, most of the material, including many of the fundamental ideas, is scatteredthroughout hundreds of research articles which are not always easily readable fornon-specialists. One of the main purposes of this book is thus to give an up-to-dateand, as much as possible, self-contained account of the most important results andideas of the field. In particular we try to find a balance between fundamental ideasand current research. Special effort is made to describe in a pedagogical way themain methods and techniques used in the study of these problems and to clar-ify the connections between several important results. Moreover, a number of theoriginal proofs have been significantly simplified. In this way, the topic should beaccessible to a larger audience of non-specialists.

The book contains five chapters. The first two are intended to be an introductionto the field and to enable the reader to get acquainted with the main ideas bystudying simple model problems, respectively of elliptic and parabolic type. Thesemodel problems are of the form

−∆u = f(u), x ∈ Ω,

u = 0, x ∈ ∂Ω,

(0.1)

x Introduction

andut −∆u = f(u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

(0.2)

where Ω ⊂ Rn and f is a superlinear function, typically f(u) = |u|p−1u for some

p > 1. The subsequent three chapters are devoted to problems with more com-plex structure; namely, elliptic and parabolic systems, equations with gradientdepending nonlinearities, and nonlocal equations. They include several problemsarising in biological or physical contexts. These chapters contain many develop-ments which reflect several aspects of current research. Although the techniquesintroduced in Chapters I and II provide efficient tools to attack some aspects ofthese problems, they often display new phenomena and specifically different be-haviors, whose study requires new ideas. Many open problems are mentioned andcommented.

For the reader’s convenience we have collected a number of frequently used re-sults in several appendices. These include estimates of solutions of linear ellipticand parabolic equations, maximum principles, and basic notions from dynamicalsystems. Also, in one of the appendices, we give an account of the local theory ofsemilinear parabolic problems based on the abstract framework of interpolation-extrapolation spaces. However, this material is not essential for the understandingof the main contents of the book and can be left for a second reading. In particular,for the case of the model problem (0.2), the most useful results on local existence-uniqueness are proved by more elementary methods in the main text. On the otherhand, we assume knowledge of the fundamentals of ordinary differential equations,of measure theory, of functional analysis (distributions, self-adjoint and compactoperators in Hilbert spaces, Sobolev-Slobodeckii spaces and their embeddings, in-terpolation, Nemytskii mapping) and of the calculus of variations (minimizing ofcoercive, weakly lower semicontinuous functionals). Finally, a section of method-ological notes and an index are provided.

We would like to stress that, due to the broadness of the field of superlinearproblems, our list of results and methods is of course not complete and is influencedin part by the interests of the authors. For reasons of space, many interesting top-ics and results could not be mentioned in this book (and we also apologize for anyomission.) In particular, we do not touch degenerate problems with superlinearsource (involving for instance porous medium, fast diffusion, or p-Laplace opera-tors), nor higher order equations (where the maximum principle does not generallyapply). We do not consider superlinear problems involving nonlinear boundary con-ditions, nor parabolic systems with convection (chemotaxis, Navier-Stokes). Theseare very interesting and intensively studied topics, but would require a book ontheir own. Finally, let us mention that there exist several textbooks and mono-graphs dealing, at least in part, with certain aspects of superlinear problems; see[460], [466], [63], [113], [405], [504], [372], [222], for example.

Introduction xi

We would like to express our gratitude to several colleagues for their carefuland critical reading of (some parts of) the manuscript, particularly H. Amann,M. Balabane, M. Chipot, M. Fila, Ph. Laurencot, P. Polacik, A. Rodrıguez-Bernal,J. Rossi, F.B. Weissler and M. Winkler. Our special thanks go to H. Amann forhis stimulating encouragements to this project. We also thank T. Hempfling fromBirkhauser for his helpfulness and the first author thanks the Slovak Literary Fundfor providing financial support.

1. Preliminaries

General

We denote by BR(x) or B(x, R) the open ball in Rn with center x and radius R.

We set BR := BR(0). The (n − 1)-dimensional unit sphere is denoted by Sn−1.The characteristic function of a given set M is denoted by χM . We write D′ ⊂⊂ Dfor D′, D ⊂ R

n if the closure of D′ is a compact subset of D. For any real numbers, we set s+ := max(s, 0) and s− := max(−s, 0). We also denote R+ := [0,∞).

Domains

Let Ω be a domain, i.e. a nonempty, connected, open subset of Rn and let k ∈ N.

We shall say that Ω is uniformly regular of class Ck (cf. [13, p. 642]), if eitherΩ = R

n or there exists a countable family (Uj , ϕj), j = 1, 2, . . . of coordinatecharts with the following properties:

(i) Each ϕj is a Ck-diffeomorphism of Uj onto the open unit ball B1 in Rn

mapping Uj∩Ω onto the “upper half-ball” B1∩(Rn−1×(0,∞)) and Uj∩∂Ωonto the flat part B1 ∩ (Rn−1 × 0). In addition, the functions ϕj and thederivatives of ϕj and ϕ−1

j up to the order k are uniformly bounded on Uj

and B1, respectively.(ii) The set

⋃j ϕ−1

j (B1/2) contains an ε-neighborhood of ∂Ω in Ω for some ε > 0.

(iii) There exists k0 ∈ N such that any k0 + 1 distinct sets Uj have an emptyintersection.

In an analogous way we define a uniformly regular domain of class C2+α (shortlydomain of class C2+α). Unless explicitly stated otherwise1, we will always assumethat

Ω ⊂ Rn is a uniformly regular domain of class C2+α for some α ∈ (0, 1).

On the other hand, we do not assume Ω to be bounded unless this is explicitlymentioned.

We denote the distance to the boundary function by

δ(x) := dist (x, ∂Ω).

The exterior unit normal on ∂Ω at a point x ∈ ∂Ω is denoted by ν(x), and theouter normal derivative by ∂ν or ∂/∂ν. The surface measure (on e.g. ∂Ω or Sn−1)will be denoted by dσ or dω.

For a given domain Ω and 0 < T < ∞, we set

QT : = Ω× (0, T ),

ST : = ∂Ω× (0, T ) (lateral boundary),

PT : = ST ∪(Ω× 0

)(parabolic boundary).

1In fact, if we want to allow nonsmooth domains, we will refer to an arbitrary domain.

2 1. Preliminaries

Functions of space and time

Let u = u(x, t) be a real function of the space variable x ∈ Ω and the time variablet. Without fearing confusion we will also consider u as a function of a single variablet with values in a space of functions defined in Ω, hence u(t)(x) = u(x, t).

By a solution of a PDE being positive we usually mean that u(x) > 0 oru(x, t) > 0 in the domain under consideration. Note that, due to the strong max-imum principles in Appendix F, positive is often equivalent to nontrivial nonneg-ative.

Radial functions. We say that a domain Ω ⊂ Rn is symmetric if either Ω = R

n,or Ω = BR = x ∈ R

n : |x| < R, or Ω = x ∈ Rn : R < |x| < R′, where

0 < R < R′ ≤ ∞ (an annulus if R′ < ∞). Denote r = |x| and let J ⊂ R bean interval. A function u defined on a symmetric domain Ω (resp., on Ω × J) issaid to be radially symmetric, or simply radial, if it can be written in the formu = u(r) (resp., u = u(r, t) for each t ∈ J). The function u is said to be radialnonincreasing if it is radial and if, moreover, u is nonincreasing as a functionof r.

Banach spaces and linear operators

If X is a Banach space and p ≥ 1, then X ′ and p′ denote the (topological) dualspace and dual exponent (1/p + 1/p′ = 1), respectively. We write X → Y orX →→ Y if X is continuously or compactly embedded in Y , respectively. If bothX → Y and Y → X (that is X and Y coincide and carry equivalent norms), thenwe write X

.= Y . We denote by L(X, Y ) the space of continuous linear operatorsA : X → Y , L(X) = L(X, X). If A is a linear operator in X with the domain ofdefinition D(A) and Y ⊂ X , then the operator AY , the Y -realization of A, isdefined by AY u = Au, D(AY ) := u ∈ D(A) ∩ Y : Au ∈ Y .

Function spaces

We denote by D(Ω) the space of C∞-functions with compact support in Ω. Thenorms in the Sobolev space W k,p(Ω) (or the Sobolev-Slobodeckii space W k,p(Ω)if k is not an integer) and the Lebesgue space Lp(Ω) will be denoted by ‖ · ‖k,p

and ‖ · ‖p, respectively. We denote by W 1,20 (Ω) the closure of D(Ω) in W 1,2(Ω).

The spaces W k,2(Ω), k ∈ N, and W 1,20 (Ω) will also be denoted as Hk(Ω) and

H10 (Ω), respectively. The functions in these spaces are usually understood to be real

valued. If no confusion is likely, we shall use the same notation for similar spacesof functions with values in R

n. Otherwise we shall use the notation Lp(Ω, Rn), forexample.

Let Ω be a bounded domain in Rn (not necessarily smooth). The weighted

Lebesgue spaces Lpδ(Ω) are defined as follows. Denoting as before

δ(x) = dist(x, ∂Ω), x ∈ Ω,

1. Preliminaries 3

we put, for all 1 ≤ p ≤ ∞,

Lpδ = Lp

δ(Ω) := Lp(Ω; δ(x) dx).

For 1 ≤ p < ∞, Lpδ is endowed with the norm

‖u‖p,δ =(∫

Ω

|u(x)|p δ(x) dx)1/p

.

Remark 1.1. Let us note that L∞δ (Ω) = L∞(Ω), with same norm. Indeed, L∞

δ (Ω)consists, by definition, of those measurable functions that are essentially boundedwith respect to the measure δ(x) dx.

For any 1 ≤ p < ∞, the uniformly local Lebesgue space (cf. [297], [253]) Lpul is

defined byLp

ul = Lpul(R

n) =φ ∈ Lp

loc(Rn) : ‖φ‖p,ul <∞

,

where

‖φ‖p,ul := supa∈Rn

(∫|y−a|<1

|φ(y)|p dy)1/p

.

These are Banach spaces with the norm ‖.‖p,ul. Also, for p =∞, we define L∞ul :=

L∞ = L∞(Rn). We note that Lrul → Lp

ul whenever 1 ≤ p ≤ r ≤ ∞.In what follows X denotes a Banach space.Let M be a metric space. Then B(M, X), BC(M, X), BUC(M, X) denote

the Banach spaces of bounded, bounded and continuous, bounded and uniformlycontinuous functions u : M → X , respectively, all endowed with the sup-norm

‖u‖∞ = ‖u‖∞,M := supt∈M

‖u(t)‖X .

We denote by C(M, X) the space of continuous functions endowed with the topol-ogy of locally uniform convergence. If M is locally compact, then we denote byC0(M, X) the space of functions u ∈ BUC(M, X) with the following property:Given ε > 0, there exists a compact set K ⊂ M such that ‖u(t)‖X < ε for allt ∈ M \K. We also set B(M) := B(M, R), BC(M) := BC(M, R), etc.

Let M ⊂ Rn. A function u : M → X is said to be locally Holder continuous if,

for each point t ∈ M , there exist α ∈ (0, 1), C > 0 and a neighborhood V of t,such that

uα,M∩V := supx,y∈M∩V, x =y

‖u(x)− u(y)‖X

|x− y|α <∞. (1.1)

If α in (1.1) can be chosen independent of t ∈ M , then u is said to be locallyα-Holder continuous. The space of such functions is denoted by Cα(M, X) (orCα(M) if X = R) and endowed with the family of seminorms ‖ · ‖∞,K + ·α,K ,

4 1. Preliminaries

where K runs over all compact subsets of M . By UCα(M, X), α ∈ (0, 1), wedenote the set of functions u : M → X such that

uα := uα,M < ∞.

The norm in the Banach space BUCα(M, X) = B(M, X) ∩ UCα(M, X) is thesum of the sup-norm and the seminorm ·α. Note that if M is compact, then anylocally Holder continuous function u : M → X belongs to BUCα(M, X) for someα and Cα(M, X) = BUCα(M, X).

If Ω is an arbitrary domain in Rn, then BC1(Ω) denotes the space of functions

u ∈ BC(Ω) whose first derivatives in Ω are bounded, continuous and can becontinuously extended to Ω. The norm of a function u in this space is definedas the sum of sup-norms of u and its first-order derivatives. The spaces BCk(Ω)and BUCk(Ω), k ≥ 1 integer, are defined in an obvious way. If no confusion islikely, we shall denote their norms by ‖ · ‖BCk . The spaces Ck+α(Ω), UCk+α(Ω),BUCk+α(Ω), where k ≥ 1 is an integer and α ∈ (0, 1) are defined similarly.

Let Ω be a bounded domain in Rn. Then Ω is compact, hence any function in

C(Ω) is bounded and uniformly continuous. On the other hand, the functions inBUC(Ω) can be uniquely extended to functions in C(Ω). Identifying the functionu ∈ BUC(Ω) with its extension and endowing the space C(Ω) with the sup-norm,we can write BUC(Ω) = C(Ω). Similarly, BUCα(Ω) = Cα(Ω).

If Q ⊂ Rn × R is a domain in space and time, then C2,1(Q) is the space

of functions which are twice continuously differentiable in the spatial variable xand once in the time variable t. The space BC2,1(Q) has obvious meaning. Ifu ∈ Lp(Q), then ut, Dxu and D2

xu denote the time derivative and first and secondspatial derivatives of u in the sense of distributions. Alternatively, we shall alsouse the notation ∇u, D2u instead of Dxu, D2

xu. We denote by W 2,1;p(Q) the spaceof functions u ∈ Lp(Q) satisfying ut, Dxu, D2

xu ∈ Lp(Q), endowed with the norm

‖u‖2,1;p = ‖u‖2,1;p;Q := ‖u‖p;Q + ‖Dxu‖p;Q + ‖D2xu‖p;Q + ‖ut‖p;Q.

Let Q = QT = Ω× (0, T ) where Ω is an arbitrary domain in Rn and T > 0. Given

α ∈ (0, 1] set

[f ]α;Q = sup |f(x, t)− f(y, s)||x− y|α + |t− s|α/2

: x, y ∈ Ω, t, s ∈ (0, T ), (x, t) = (y, s).

Let k be a nonnegative integer, α ∈ (0, 1) and a = k + α. Then we put

|f |a;Q =∑

|β|+2j≤k

supQ|Dβ

xDjt f |+

∑|β|+2j=k

[DβxDj

t f ]α;Q

and BUCa,a/2(Q) := f : |f |a;Q <∞. The spaces UCa,a/2(Q) and Ca,a/2(Q) aredefined analogously as in the case of functions defined in Rn. Note that if p > n+2,

1. Preliminaries 5

a < 2 − (n + 2)/p and Ω is smooth enough (for example, if Ω satisfies a uniforminterior cone condition), then

W 2,1;p(Q) → BUCa,a/2(Q); (1.2)

see [320, Lemmas II.3.3, II.3.4], [399, Theorem 6.9] and the references therein forthis statement and more general embedding and trace theorems for anisotropicspaces. Embedding (1.2) can also be derived by using the interpolation embeddingin Proposition 51.3 and embeddings for isotropic spaces.

Eigenvalues and eigenfunctions

If Ω is bounded, then we denote by λ1, λ2, . . . the eigenvalues of−∆ in W 1,20 (Ω) and

by ϕ1, ϕ2, . . . the corresponding eigenfunctions. Recall that λ1 < λ2 ≤ λ3 ≤ · · · ,λk →∞ as k →∞, that

1λ1

= sup∫

Ω

u2 dx : u ∈W 1,20 (Ω),

∫Ω

|∇u|2 dx = 1, (1.3)

and that we can choose ϕ1 > 0. Unless explicitly stated otherwise, we shall assumethat ϕ1 is normalized by ∫

Ω

ϕ1 dx = 1.

We shall often use the fact that if Ω is of class C2, then there exist constantsc1, c2 > 0 such that

c1δ(x) ≤ ϕ1(x) ≤ c2δ(x), x ∈ Ω (1.4)

(this is a consequence of u ∈ C1(Ω) and of Hopf’s lemma; cf. Proposition 52.1(iii)).

Further frequent notation

We denote by G(x, y, t) = GΩ(x, y, t) the Dirichlet heat kernel; Gt(x) = G(x, t) isthe Gaussian heat kernel in R

n. The (elliptic) Dirichlet Green kernel is denotedby K(x, y) = KΩ(x, y). We implicitly mean by e−tA the Dirichlet heat semigroupin Ω.

The Dirac distribution at point y will be denoted by δy.We shall use the symbols C, C1, etc. to denote various positive constants. The

dependence of these constants will be made precise whenever necessary.Definitions of various critical exponents (pF , pBT , psg, pS , pJL, pL, 2∗, 2∗, qc) and

other symbols can be found via the List of Symbols.

Chapter I

Model Elliptic Problems

2. Introduction

In Chapter I, we study the problem

−∆u = f(x, u), x ∈ Ω,

u = 0, x ∈ ∂Ω,

(2.1)

where f : Ω×R → R is a Caratheodory function (i.e. f(·, u) is measurable for anyu ∈ R and f(x, ·) is continuous for a.e. x ∈ Ω). Of course, the boundary conditionin (2.1) is not present if Ω = R

n. We will be mainly interested in the model case

f(x, u) = |u|p−1u + λu, where p > 1 and λ ∈ R. (2.2)

Denote by pS the critical Sobolev exponent,

pS := ∞ if n ≤ 2,

(n + 2)/(n− 2) if n > 2.

We shall refer to the cases p < pS , p = pS or p > pS as to (Sobolev) subcritical,critical or supercritical, respectively.

3. Classical and weak solutions

Let u be a solution of (2.1) and f(x) := f(x, u(x)). Then u solves the linear

problem−∆u = f in Ω,

u = 0 on ∂Ω.

(3.1)

In what follows we define several types of solutions of the linear problem (3.1)(and, consequently, of (2.1)).

Definition 3.1. (i) We call u a classical solution of (3.1) if f ∈ C(Ω), u ∈C2(Ω) ∩ C(Ω) and u satisfies the equation and the boundary condition in (3.1)pointwise.

8 I. Model Elliptic Problems

(ii) We call u ∈W 1,20 (Ω) a variational solution of (3.1) if f ∈

(W 1,2

0 (Ω))′ and∫

Ω

∇u · ∇ϕdx =∫

Ω

fϕ dx for all ϕ ∈ W 1,20 (Ω). (3.2)

(iii) Let Ω be bounded, u ∈ L1(Ω). Set

δ(x) := dist (x, ∂Ω) and L1δ(Ω) := L1(Ω, δ(x)dx).

We call u an L1-solution of (3.1) if f ∈ L1(Ω) and∫Ω

u(−∆ϕ) dx =∫

Ω

fϕ dx for all ϕ ∈ C2(Ω), ϕ = 0 on ∂Ω. (3.3)

More generally, we call u an L1δ-solution, or a very weak solution, of (3.1)

if f ∈ L1δ(Ω) and (3.3) is satisfied. Note that the definition makes sense since

|ϕ| ≤ Cδ hence fϕ ∈ L1(Ω). Existence-uniqueness and properties of L1δ solutions

of the linear problem (3.1) are studied in Appendix C.(iv) If Ω = R

n, then u ∈ L1loc(Ω) is called a distributional solution of (3.1) if

the integral identity in (3.3) is true for all ϕ ∈ D(Rn).

Remarks 3.2. (i) If we assume that f is a bounded Radon measure in Ω (insteadof f ∈ L1(Ω)), then the definition of an L1-solution still makes sense and we referto [20] and the references therein for properties of such solutions.

(ii) If f ∈ L∞(Ω), then any classical solution of (3.1) satisfies u ∈ W 2,q(K) forany K ⊂⊂ Ω and any q < ∞. This is a consequence of Remark 47.4(iii). If wefurther assume that f is locally Holder continuous in Ω, then u ∈ C2(Ω).

(iii) Assume Ω bounded. If f ∈ C(Ω), for example, then any classical solution of(3.1) is also a variational solution (this follows from Remark (ii) and integration byparts). If f ∈ L2(Ω), then any variational solution is an L1-solution. Some otherrelations between various types of solutions defined above will be mentioned below(see also Lemma 47.7 in Appendix A).

In the following sections we shall often use variational methods in order to provethe solvability of (2.1). Therefore, we derive now a sufficient condition on f whichguarantees that any variational solution of (2.1) is classical.

If n ≥ 3 we set 2∗ := pS + 1 = 2n/(n − 2), 2∗ := (2∗)′ = 2n/(n + 2). Assumethat the Caratheodory function f satisfies the following growth assumption

|f(x, u)| ≤ α(x)+Cf (|u|+|u|p), α ∈ L(p+1)′(Ω)+L2(Ω), Cf > 0, p ≤ pS . (3.4)

This growth condition can be significantly weakened if n ≤ 2 but (3.4) will besufficient for our purposes; cf. (2.2). Denote

F (x, u) :=∫ u

0

f(x, s) ds

3. Classical and weak solutions 9

andE(u) :=

12

∫Ω

|∇u(x)|2 dx−∫

Ω

F(x, u(x)

)dx. (3.5)

Since p ≤ pS we have W 1,2(Ω) → Lp+1(Ω) and the embedding is compact providedp < pS and Ω is bounded. In addition, the energy functional E is C1 (continuouslyFrechet differentiable) in W 1,2(Ω) and

E′(u)ϕ =∫

Ω

∇u · ∇ϕ dx−∫

Ω

f(·, u)ϕdx

for all u, ϕ ∈ W 1,2(Ω). In particular, each critical point of E in W 1,20 (Ω) is a

variational solution of (2.1).The following proposition is essentially due to [96]; our proof closely follows the

proof of [505, Lemma B.3].

Proposition 3.3. Assume (3.4). If n ≥ 3 assume also α ∈ Ln/2(Ω). Let u be avariational solution of (2.1). Then u ∈ Lq(Ω) for all q ∈ [2,∞).

Proof. Since the assertion is obviously true if n ≤ 2 due to W 1,2(Ω) → Lq(Ω),we may assume n ≥ 3.

Denote f(x) := f(x, u(x)

). Then

|f | ≤ α + Cf (|u|+ |u|p) ≤ a + b + 2Cf (|u|+ |u|pS ),

where a := αχ|u|>1 ∈ Ln/2(Ω), b := αχ|u|≤1 and α can be written in the formα = α1 + α2 with α1 ∈ L(p+1)′(Ω), α2 ∈ L2(Ω).

Choose s ≥ 0 such that u ∈ L2(s+1)(Ω). We shall prove that u ∈ L2∗(s+1)(Ω) sothat an obvious bootstrap argument proves the assertion.

Choose L > 0 and setψ := min

(|u|s, L

), ϕ := uψ2, ΩL := x ∈ Ω : |u|s ≤ L.

In what follows we denote by C, C1, C2 various positive constants which may varyfrom step to step and which may depend on u, s, α, Cf but which are independentof L. We have

∇(uψ) = (1 + sχΩL)(∇u)ψ,

∇ϕ = (1 + 2sχΩL)(∇u)ψ2,

and ϕ ∈W 1,20 (Ω). Therefore, we obtain∫

Ω

|∇u|2ψ2 dx ≤∫

Ω

∇u · ∇ϕdx =∫

Ω

fϕ dx =∫

Ω

fuψ2 dx

≤ C

∫Ω

[(a + b)|u|ψ2 + u2ψ2 + |u|2∗

ψ2]dx

≤ C

∫Ω

[au2ψ2 + b|u|+ |u|2s+2 + |u|2∗

ψ2]dx

≤ C(1 +

∫Ω

(a + |u|2∗−2)u2ψ2 dx),

where we have used∫Ω

b|u| dx ≤∫

Ω

α|u| dx ≤∫

Ω

(|α1|+ |α2|)|u| dx

≤ ‖α1‖(p+1)′‖u‖p+1 + ‖α2‖2‖u‖2 = C.

Consequently, denoting v := a + |u|2∗−2 ∈ Ln/2(Ω), we obtain∫Ω

|∇(uψ)|2 dx ≤ C

∫Ω

|∇u|2ψ2 dx ≤ C(1 +

∫Ω

vu2ψ2 dx)

≤ C(1 + K

∫|v|≤K

u2ψ2 dx +∫|v|>K

v(uψ)2 dx)

≤ C(1 + K

∫Ω

|u|2s+2 dx +(∫

|v|>K

vn/2 dx)2/n(∫

Ω

|uψ|2∗dx)(n−2)/n)

≤ C1(1 + K) + C2εK

∫Ω

|∇(uψ)|2 dx,

where εK :=(∫

|v|>K vn/2 dx)2/n→ 0 as K → +∞. Choosing K such that C2εK <

1/2 we arrive at∫ΩL

|∇(|u|s+1)|2 dx =∫

ΩL

|∇(uψ)|2 dx ≤ 2C1(1 + K).

Letting L → +∞ we get |u|s+1 ∈W 1,2(Ω), hence u ∈ L2∗(s+1)(Ω).

Corollary 3.4. If f has the form (2.2) with p ≤ pS, then any variational solutionu of (2.1) is also a classical solution. Moreover, u ∈ C2(Ω).

Proof. The assertion is a consequence of standard regularity results for linearelliptic equations. More precisely, for any 2 ≤ q < ∞, since f := f(u) ∈ Lq(Ω),Theorem 47.3(i) implies the existence of u ∈W 2,q ∩W 1,q

0 (Ω) such that −∆u = f .Since u, u ∈ H1

0 (Ω), the maximum principle in Proposition 52.3(i) yields u = u.Due to the embedding W 2,q(Ω) ⊂ C1(Ω) for q > n, we deduce that f ∈ C1(Ω).Applying now Theorem 47.3(ii), and Proposition 52.3(i) again, we deduce thatu ∈ C2(Ω).

As for L1-solutions, we have the following regularity result (we shall see inRemarks 3.6 below that the growth conditions in Propositions 3.3 and 3.5 areoptimal).

Proposition 3.5. Assume Ω bounded. Let the Caratheodory function f satisfythe growth assumption

|f(x, u)| ≤ C(1 + |u|p), p < psg, (3.6)

3. Classical and weak solutions 11

where psg is defined in (3.8). Let u be an L1-solution of (2.1). Then u ∈ C0 ∩W 2,q(Ω) for all finite q.

Proof. It is based on a simple bootstrap argument. Fix ρ ∈ (1, n/(n− 2)p) andput f(x) = f

(x, u(x)

). Assume that there holds

f ∈ Lρi

(Ω) (3.7)

for some i ≥ 0 (this is true for i = 0 by assumption). Since

1ρi− 1

pρi+1=

1ρi

(1− 1

pρ

)<

2n

,

by using Proposition 47.5(i), we obtain u ∈ Lpρi+1(Ω), hence f ∈ Lρi+1

(Ω) dueto (3.6). By induction, it follows that (3.7) is true for all integers i. In particularf ∈ Lk(Ω) for some k > n/2 and we may apply Proposition 47.5(i) once moreto deduce that u ∈ L∞(Ω). The conclusion then follows similarly as in the proofof Corollary 3.4 (using the uniqueness part of Theorem 49.1 instead of Proposi-tion 52.3).

Remarks 3.6. (i) Singular solution. Define the exponent

psg := ∞ if n ≤ 2,

n/(n− 2) if n > 2.(3.8)

For p > psg (hence n ≥ 3), we let

U∗(r) := cpr−2/(p−1), r > 0, where cp−1

p :=2

(p− 1)2((n− 2)p− n

). (3.9)

One can easily check that u∗(x) := U∗(|x|) is a positive, radial distributional solu-tion of the equation −∆u = up in R

n. This singular solution (hence the notationpsg) plays an important role in the study of the parabolic problem (0.2) withf(u) = |u|p−1u (see for example Theorems 20.5, 22.4 and 23.10).

On the other hand, if we set u(x) := u∗(x) − cp for 0 < |x| ≤ 1, Ω := B1(0) =x ∈ R

n : |x| < 1, then it is easy to verify that u is an L1-solution of (2.1) withf(x, u) = (u+ cp)p. Moreover, u is a variational solution of this problem if p > pS .Hence the condition p ≤ pS in Proposition 3.3 is necessary.

(ii) Let n ≥ 3 and let Ω be bounded, f ∈ C1, |f(x, u)| ≤ C(1 + |u|p). Theexample in (i) shows that an L1-solution need not be classical if p > psg. In fact,it was proved in [44], [394] that the problem

−∆u = |u|p−1u in Ω,

u = 0 on ∂Ω,

(3.10)

has a positive unbounded radial L1-solution u ∈ C2(Ω\0) provided p ∈ [psg, pS)and Ω = B1(0). See also [404] and the references therein for related nonradialresults.

(iii) For the case of L1δ-solutions, we shall see in Section 11 that the critical

exponent is different, namely (n + 1)/(n− 1).

Remark 3.7. Classical vs. very weak solutions for the nonlinear eigen-value problem. Another type of relations between different notions of solutionsappears when one considers the nonlinear eigenvalue problem

−∆u = λf(u), x ∈ Ω,

u = 0, x ∈ ∂Ω.

(3.11)

Here we assume that f : [0,∞) → (0,∞) is a C1 nondecreasing, convex function,and λ > 0. Namely, it was proved in [94] (see also [233] for earlier related results)that if there exists a very weak solution of (3.11) for some λ0 > 0, then thereexists a classical solution for all λ ∈ (0, λ0). The proof is based on a perturbationargument relying on a variant of Lemma 27.4 below. As a consequence of this andof results from [305], [142], assuming in addition that limu→∞ f(u)/u = ∞, thereexists λ∗ ∈ (0,∞) such that:

(i) for 0 < λ < λ∗, problem (3.11) has a (unique minimal) classical solutionuλ, and the map λ → uλ is increasing;

(ii) for λ = λ∗, problem (3.11) has a very weak solution defined by uλ∗ =limλ↑λ∗ uλ;

(iii) for λ > λ∗, problem (3.11) has no very weak solution.On the other hand, the solution uλ∗ may be classical or singular, depending on

the nonlinearity. For instance, in the case f(u) = (u + 1)p with Ω = BR, (3.11)has a classical solution for λ = λ∗ if and only if p < pJL, where pJL is defined in(9.3); in the case f(u) = eu, the condition is replaced with n ≤ 9 (see [293], [369]).Illustrations of these facts appear on the bifurcation diagram in Remark 6.10(ii)(see Figure 3).

4. Isolated singularities

In this section we study the question of isolated singularities of positive classicalsolutions to the equation −∆u = up. The following result classifies the possiblesingular behaviors for subcritical or critical p.

Theorem 4.1. Let n ≥ 3 and 1 < p ≤ pS. Assume that u is a positive classicalsolution of

−∆u = up in B1 \ 0 (4.1)

4. Isolated singularities 13

and that u is unbounded at 0. Then there exist constants C2 ≥ C1 > 0 such that

C1ψ(x) ≤ u(x) ≤ C2ψ(x), 0 < |x| < 1/2,

where

ψ(x) =

⎧⎪⎨⎪⎩|x|2−n if 1 0.

Furthermore, for all p > 1, we have the following result, which explains in whatsense the equation can be extended to the whole unit ball.

Theorem 4.2. Let p > 1 and n ≥ 3. Assume that u is a positive classical solutionof

−∆u = up in B1 \ 0.

(i) Then up ∈ L1loc(B1) and there exists a ≥ 0 such that u is a solution of

−∆u = up + aδ0 in D′(B1),

where δ0 denotes the Dirac delta distribution. Moreover, we have a ≤ a with a =a(n, p) > 0.(ii) If p < psg and a = 0, then the singularity is removable, i.e. u is bounded nearx = 0.(iii) If p ≥ psg, then a = 0.

Remarks 4.3. (i) Theorem 4.1 follows from [340], [44], [240] (see also [83]), and[108], for the cases p < psg, p = psg, psg < p < pS and p = pS respectively.Theorem 4.2 follows from [97] and [340]. See also the book [523] for further resultsand references.

(ii) Under the assumptions of Theorem 4.1 with psg 0, as x → 0(see the proof below). Examples in [340] show that singular solutions do exist for1 pS , then the upper estimate u(x) ≤ C|x|−2/(p−1) is still true in theradial case (cf. [213], [397]). In fact, as a consequence of −(rn−1ur)r = rn−1up > 0,for r > 0 small, we have either ur > 0, hence u bounded, or ur ≤ 0. In this secondcase, by integration, we get −ur ≥ r1−n

∫ r

0 sn−1up(s) ds ≥ (r/n)up for r > 0 small,hence (u1−p)r ≥ Cr, and the upper estimate follows by a further integration. Theestimate is unknown in the nonradial case for p > pS , but related integral estimatesof solutions can be found in e.g. [238] and [89].

(iv) A result similar to Theorem 4.1 is true for n = 2, with ψ(x) given by thefundamental solution log |x| instead of |x|2−n. These results are related to the factthat the (H1–) capacity of a point is 0 when n ≥ 2. When the origin is replaced bya closed subset of 0 capacity, related results can be found in [170]. On the otherhand, the upper estimate u(x) ≤ C|x|−2/(p−1), from the case psg < p < pS , can begeneralized to sets other than a single point (see Theorem 8.7 in Section 8).

We shall first prove Theorem 4.2. Theorem 4.1 in the case 1 < p < psg willthen follow as a consequence of Theorem 4.2 and of a bootstrap argument. In thecase psg < p < pS , the upper estimate will be a consequence of the more generalresult Theorem 8.7 in Section 8. For the cases p = psg, p = pS , and for the lowerestimate when psg < p < pS , see the above mentioned references.

In view of the proofs, we introduce the following notation. We denote by Γ(x) =cn|x|2−n the fundamental solution of the Laplacian (Newton potential), i.e. −∆Γ =δ0 in D′(Rn). We let ω = x ∈ R

n : |x| < 1/2 and fix χ ∈ D(B1) such that χ = 1on ω and 0 ≤ χ ≤ 1. For each positive integer j, denote χj(x) = χ(jx). By astraightforward calculation using n ≥ 3, we see that χj → 0 in H1(B1) as j →∞.For any ϕ ∈ D(B1), we put ϕj := (1− χj)ϕ. Observe that ϕj → ϕ in H1(B1).

We need the following lemma.

Lemma 4.4. Let n ≥ 3. Assume that u ∈ C2(B1 \ 0) satisfies u ≥ 0 and

−∆u ≥ 0 in B1 \ 0.Then u ∈ L1

loc(B1) and−∆u ≥ 0 in D′(B1).

Proof. For each k > 0, we take a function Gk ∈ C2([0,∞)) such that Gk(s) = sfor 0 ≤ s ≤ k, Gk(s) = k + 1 for s large, G′

k ≥ 0 and G′′k ≤ 0. Define uk := Gk(u)

and note that the sequence ukk is monotone increasing and converges to upointwise in B1 \ 0. The function uk satisfies

−∆uk = −G′k(u)∆u −G′′

k(u)|∇u|2 ≥ 0 in B1 \ 0. (4.2)

Fix α > 0 and ϕ ∈ D(B1). Multiplying inequality (4.2) by the test-functionϕ2

j (1 + uk)−α and integrating by parts, we obtain

0 ≤∫

B1

∇uk · ∇(ϕ2j (1 + uk)−α)

= −α

∫B1

|∇uk|2ϕ2j (1 + uk)−1−α + 2

∫B1

∇uk · ∇ϕj(1 + uk)−αϕj .

It follows that

α

∫B1

|∇uk|2ϕ2j (1 + uk)−1−α

≤ α

2

∫B1

|∇uk|2ϕ2j(1 + uk)−1−α + C(α)

∫B1

|∇ϕj |2(1 + uk)1−α,

hence ∫B1

|∇uk|2ϕ2j(1 + uk)−1−α ≤ C(α)

∫B1

|∇ϕj |2(1 + uk)1−α.

Since |∇ϕj |2 → |∇ϕ|2 in L1(B1) and (1 + uk)1−α ∈ L∞(B1), we may pass to thelimit j →∞ (using Fatou’s lemma on the LHS) and we obtain∫

B1

|∇uk|2ϕ2(1 + uk)−1−α ≤ C(α)∫

B1

|∇ϕ|2(1 + uk)1−α.

First taking α = 1 and using 1 + uk ≤ k + 2, we deduce that uk ∈ H1(ω), henceuk ∈ H1

loc(B1).Next take α = 2/n. Consider ϕ such that ϕ = 1 for |x| ≤ 1/4 and with support

in ω. Applying the Sobolev and Holder inequalities, we get, for any ρ ∈ (0, 1/2),

(∫|x|<1/4

(1 + uk))n−2

n ≤ C

∫|x|<1/4

∣∣∇[(1 + uk)n−22n

]∣∣2 + C

∫|x|<1/4

(1 + uk)n−2

n

≤ C

∫ω

(1 + uk)n−2

n

≤ C

∫ρ<|x|<1/2

(1 + uk)n−2

n + Cρ2(∫

|x|≤ρ

(1 + uk))n−2

n

.

Since u is bounded on ρ < |x| < 1/2 and uk ≤ u, by taking ρ ∈ (0, 1/4) smallenough, we deduce that

∫|x|<1/4 uk ≤ C independent of k. Consequently u ∈ L1(ω),

hence u ∈ L1loc(B1), and uk → u in L1

loc(B1).Now assuming ϕ ≥ 0, we multiply inequality (4.2) by ϕj and integrate by parts.

We obtain ∫B1

∇uk · ∇ϕj =∫

B1

(−∆uk)ϕj ≥ 0.

Since uk ∈ H1loc(B1), we may pass to the limit j → ∞ to get

∫B1∇uk · ∇ϕ ≥ 0,

hence∫

B1(−∆ϕ)uk ≥ 0. Since uk → u in L1

loc(B1), we conclude that∫

B1(−∆ϕ)u ≥

0 and the proof of the lemma is complete.

Proof of Theorem 4.2. (i) By Lemma 4.4, we know that u ∈ L1loc(B1) and that

−∆u ≥ 0 in D′(B1). It follows that ∆u is a Radon measure (in other words, a0-order distribution) on ω. Indeed, for each ϕ ∈ D(B1) with supp(ϕ) ⊂ ω, using‖ϕ‖∞χ± ϕ ≥ 0, we obtain

〈−∆u, ‖ϕ‖∞χ± ϕ〉 ≥ 0

hence|〈−∆u, ϕ〉| ≤ |〈−∆u, χ〉| ‖ϕ‖∞ =: C‖ϕ‖∞. (4.3)

We next claim that up ∈ L1loc(B1). To this end, we assume ϕ ≥ 0, we multiply

(4.1) by ϕj and we integrate by parts. We obtain∫B1

upϕj = 〈−∆u, ϕj〉 ≤ C‖ϕj‖∞ ≤ C‖ϕ‖∞,

due to (4.3), and the claim follows from Fatou’s lemma.Now a classical argument in distribution theory allows us to conclude: Denote

T = ∆u + up ∈ D′(B1) and let ϕ ∈ D(B1). Since T = 0 in D′(B1 \ 0) and(1−χj)ϕ = 0 in the neighborhood of 0, we have 〈T, (1−χj)ϕ〉 = 0. Consequently,

〈T, ϕ〉 − 〈T, χj〉ϕ(0) = 〈T, ϕχj〉 − 〈T, χj〉ϕ(0) = 〈T, (ϕ− ϕ(0))χj〉. (4.4)

But since ‖(ϕ−ϕ(0))χj‖∞ → 0 as j →∞, it follows that the LHS of (4.4) convergesto 0 as j →∞. We first deduce that = limj→∞〈T, χj〉 exists (take a ϕ such thatϕ(0) = 0). Moreover, since −∆u ≥ 0 in D′(B1), we have ≤ limj→∞

∫ω upχj = 0

by dominated convergence. Returning to (4.4), we obtain

∆u + up = −aδ0 (4.5)

with a = − ≥ 0. Now let ψ ∈ D(B1) satisfy −∆ψ ≤ µψ1/p in B1 for someµ > 0 and ψ ≥ C > 0 for |x| < 2/3 (such function is given for instance byψ(x) = exp[−(1 − 2|x|2)−1] for |x|2 < 1/2 and ψ(x) = 0 otherwise). Testingequation (4.5) with ψ, we get

aψ(0) +∫

B1

upψ = −∫

B1

u∆ψ ≤ µ

∫B1

uψ1/p ≤ 12

∫B1

upψ + C(p, n).

It follows thata +

∫|x|<2/3

up < a(n, p). (4.6)

In particular, assertion (i) is proved.For further reference, we also observe that

u ≥ aΓ− C in ω. (4.7)

To show this, we first note that v := u − aΓ satisfies −∆v = up in D′(B1). ByLemma 47.7, w := χv is an L1-solution of

−∆w = g := upχ− h in B1,

w = 0 on ∂B1,

(4.8)

where h := 2∇u ·∇χ+u∆χ ∈ L∞(B1). At this point, let us introduce the functionΘ ∈ C2(B1), Θ ≥ 0, classical solution of the problem

−∆Θ = 1 in B1,

Θ = 0 on ∂B1.

(This is the so-called “torsion” function, which will be useful as a comparisonor test-function later again.) Then w + ‖h‖∞Θ is an L1-solution of (4.8) with greplaced by g + ‖h‖∞ ≥ 0. By the maximum principle part of Theorem 49.1, wededuce that w + ‖h‖∞Θ ≥ 0, hence (4.7).

(ii) Let 1 < p < psg and assume that a = 0. We have seen that w = χu is anL1-solution of (4.8). Moreover, since χ = 1 near x = 0, we may write g = wp + h in(4.8) for some h ∈ L∞(B1). It then follows from Proposition 3.5 that w ∈ L∞(B1),hence u ∈ L∞(ω).

(iii) Assume p ≥ psg. If we had a > 0, then (4.7) would imply up ≥ C|x|−(n−2)p

as x → 0 for some C > 0. Since up ∈ L1loc(B1) due to (i), we conclude that

a = 0.

Proof of Theorem 4.1 for 1 0. Denote v0 = u, α1 = n − 2, and put v1 := u − aΓ = v0 − C1|x|−α1 .Then we have

−∆v1 = up in D′(B1).

On the other hand, an easy calculation shows that −∆(|x|−α) = C(α)|x|−α−2 inD′(B1) for all α ∈ (0, n − 2) and some C(α) > 0. Set α2 := pα1 − 2 if pα1 > 2and choose α2 ∈ (0, α1) otherwise. Notice that α2 ∈ (0, α1) = (0, n − 2) in bothcases due to pα1 < n. Since up ≤ C(v1)

p+ +C|x|−pα1 ≤ C(v1)

p+ +C|x|−α2−2, there

exists C2 > 0 such that v2 := v1 − C2|x|−α2 satisfies

−∆v2 ≤ C(v1)p+ in D′(B1).

Since (v1)p+ ≤ C(v2)

p+ + C|x|−pα2 , we can iterate this procedure and we obtain

functions vi (i = 0, 1, . . . ) satisfying vi+1 = vi −Ci+1|x|−αi+1 , with αi+1 ∈ (0, αi),and

−∆vi+1 ≤ C′i(vi)

p+ in D′(B1).

Moreover, due to 0 < a ≤ a(n, p), the constants Ci, C′i may be chosen to depend

only on n, p, i.To conclude, we apply a bootstrap argument similar to that in the proof of

Proposition 3.5: Fix ρ ∈ (1, n/(n − 2)p), let Ω1 = |x| < 2/3, and assume that(vi)+ ∈ Lpρi

loc (Ω1) for some i ≥ 0 (this is true for i = 0 in view of (4.6)). Since(−∆vi+1)+ ∈ Lρi

loc(Ω1) and

1ρi− 1

pρi+1=

1ρi

(1− 1

pρ

)<

2n

,

we may apply Proposition 47.6(ii) to deduce that (vi+1)+ ∈ Lpρi+1

loc (Ω1). By iter-ating, we get (vi)+ ∈ Lk

loc(Ω1) for some sufficiently large i and some k > n/2. We

may then apply Proposition 47.6(ii) once more to deduce that (vi+1)+ ∈ L∞(ω).This implies

u− aΓ = v1 = vi+1 +i+1∑j=2

Cj |x|−αj ≤ C(1 + |x|−α2), |x| < 1/2.

Moreover, starting from (4.6), it is easy to check that the constant C depends onlyon n, p. This along with (4.7) yields the conclusion.

5. Pohozaev’s identity and nonexistence results

In this section we prove the nonexistence of nontrivial solutions of (2.1) providedf satisfies (2.2) with p ≥ pS , λ ≤ 0 and Ω is a bounded starshaped domain. Thefollowing identity [419] plays a crucial role in the proof.

Theorem 5.1. Let u be a classical solution of (2.1) with f = f(u) being locallyLipschitz and Ω bounded. Then

n− 22

∫Ω

|∇u|2 dx− n

∫Ω

F (u) dx +12

∫∂Ω

∣∣∣∂u

∂ν

∣∣∣2x · ν dσ = 0, (5.1)

where F (u) =∫ u

0 f(s)ds.

Proof. First notice that u ∈ C2(Ω) (see Remark 3.2(ii)). Using integration byparts we obtain∫

Ω

[∇u · ∇(x · ∇u)− |∇u|2

]dx

=∫

Ω

∑i,j

∂u

∂xixj

∂

∂xi

( ∂u

∂xj

)dx =

∫Ω

∑i,j

∂u

∂xixj

∂

∂xj

( ∂u

∂xi

)dx

=∫

∂Ω

∑i,j

∂u

∂xixj

∂u

∂xiνj dσ − n

∫Ω

|∇u|2 dx−∫

Ω

∑i,j

∂

∂xj

( ∂u

∂xi

)xj

∂u

∂xidx,

hence ∫Ω

∑i,j

∂u

∂xixj

∂

∂xj

( ∂u

∂xi

)dx =

12

(∫∂Ω

∣∣∣∂u

∂ν

∣∣∣2x · ν dσ − n

∫Ω

|∇u|2 dx)

and ∫Ω

∇u · ∇(x · ∇u)dx =12

∫∂Ω

∣∣∣∂u

∂ν

∣∣∣2x · ν dσ − n− 22

∫Ω

|∇u|2 dx.

5. Pohozaev’s identity and nonexistence results 19

Multiplying the equation in (2.1) by x·∇u, integrating over Ω and denoting the left-or the right-hand side of the resulting equation by (LHS) or (RHS), respectively,we obtain

−(LHS) =∫

Ω

∆u(x · ∇u) dx =∫

∂Ω

∂u

∂ν(x · ∇u)dσ −

∫Ω

∇u · ∇(x · ∇u)dx

=12

∫∂Ω

∣∣∣∂u

∂ν

∣∣∣2x · ν dσ +n− 2

2

∫Ω

|∇u|2 dx,

(RHS) =∫

Ω

f(u)(x · ∇u) dx

=∫

∂Ω

F (u)(x · ν) dσ − n

∫Ω

F (u) dx = −n

∫Ω

F (u) dx,

where we have used that, on ∂Ω, ∇u = Cν for suitable C ∈ R and F (u) = F (0) =0. The comparison of (LHS) and (RHS) yields now the assertion.

Corollary 5.2. Assume Ω bounded and starshaped with respect to some pointx0 ∈ Ω (i.e. the segment [x0, x] is a subset of Ω for any x ∈ Ω), n ≥ 3. Assumethat

F (u) ≤ n− 22n

f(u)u for all u. (5.2)

Then (2.1) does not possess classical positive solutions. If, in addition, f(0) = 0,then (2.1) does not possess classical nontrivial solutions.

Condition (5.2) is satisfied if, for example, f(u) = |u|p−1u + λu, p ≥ pS andλ ≤ 0.

Proof. We proceed by contradiction. We can assume that Ω is starshaped withrespect to x0 = 0. Then x · ν ≥ 0 on ∂Ω and∫

∂Ω

x · ν dσ =∫

Ω

∆(x2

2

)dx > 0,

hence x · ν > 0 on a set of positive surface measure in ∂Ω.If u is a positive solution of (2.1), then ∂u/∂ν < 0 on ∂Ω by the maximum

principle and we obtain12

∫∂Ω

∣∣∣∂u

∂ν

∣∣∣2x · ν dσ > 0. (5.3)

Multiplication of the equation in (2.1) by u and integration by parts yields∫Ω

|∇u|2 dx =∫

Ω

f(u)udx (5.4).

Using (5.1), (5.3), (5.4) we arrive at∫Ω

[n− 22n

f(u)u− F (u)]dx < 0,


which contradicts (5.2).

If f(0) = 0 and u is a sign-changing solution of (2.1), then the assertion followsfrom the unique continuation property. In fact, let x1 ∈ ∂Ω be such that x · ν > 0in a neighborhood Γ1 of x1 in ∂Ω (recall that ∂Ω is smooth). Then the abovearguments guarantee ∂u/∂ν = 0 on Γ1. Since u = 0 and ∆u = f(u) = 0 on ∂Ω,all the second derivatives of u have to vanish on Γ1. Set u(x) := 0 for x /∈ Ω. Thenu is a solution of (2.1) in a neighborhood of Γ1, hence u ≡ 0 in this neighborhooddue to [272, Satz 2]. Using the same result one can easily show u ≡ 0 in Ω.

Remark 5.3. The idea of considering the multiplier x · ∇u was used before in[455] in the linear case f(u) = µu (for a different purpose, namely an integralrepresentation of the eigenvalues of the Laplacian). Identities similar to (5.1) (seeLemma 31.4 for the case of systems and see also [431]) are sometimes called Rellich-Pohozaev type identities in the literature.

6. Homogeneous nonlinearities

In this section we use variational methods in order to study the problem

−∆u = |u|p−1u + λu, x ∈ Ω,

u = 0, x ∈ ∂Ω.

(6.1)

The energy functional E has the form E(u) = Ψ(u)− Φ(u), where

Ψ(u) :=12

∫Ω

[|∇u(x)|2 − λu2

]dx and Φ(u) :=

1p + 1

∫Ω

|u|p+1 dx. (6.2)

Notice that Ψ is quadratic and Φ is positively homogeneous of order p + 1 = 2.Therefore, if

Ψ′(w) = µΦ′(w) (6.3)

for some µ > 0, then, setting t := µ1/(p−1), we get

E′(tw) = Ψ′(tw) − Φ′(tw) = t[Ψ′(w)− tp−1Φ′(w)

]= 0. (6.4)

Consequently, tw is a critical point of E, hence a classical solution of (6.1) ifp ≤ pS . A nontrivial function w satisfying (6.3) will be found by minimizing thefunctional Ψ with respect to the set M := u : Φ(u) = 1 and using the followingwell-known Lagrange multiplier rule.

6. Homogeneous nonlinearities 21

Theorem 6.1. Let X be a real Banach space, w ∈ X and let Ψ, Φ1, . . . ,Φk : X →R be C1 in a neighborhood of w. Denote M := u ∈ X : Φi(u) = Φi(w) for i =1, . . . , k and assume that w is a local minimizer of Ψ with respect to the set M .If Φ′

1(w), . . . ,Φ′k(w) are linearly independent, then there exist µ1, . . . , µk ∈ R such

that

Ψ′(w) =k∑

i=1

µiΦ′i(w).

Our proofs of the main results of this section (Theorem 6.2 and Theorem 6.7(i))follow those in [505, Theorem I.2.1 and Lemma III.2.2]. Let us first consider thesubcritical case.

Theorem 6.2. Assume Ω bounded. Let 1 0,

the functional Ψ is convex and coercive. Let uk ∈ M := u ∈ X : Φ(u) = 1,uk u in X . Then uk → u in Lp+1(Ω) due to X →→ Lp+1(Ω), hence u ∈ M .Consequently, the set M is weakly sequentially closed in the reflexive space X andthere exists w ∈M such that Ψ(w) = infM Ψ. Since |w| ∈ M and Ψ(|w|) = Ψ(w),we may assume w ≥ 0. Moreover, Φ′(w)w = (p+1)Φ(w) = p+1, hence Φ′(w) = 0.Theorem 6.1 guarantees the existence of µ ∈ R such that Ψ′(w) = µΦ′(w), hence

0 < 2Ψ(w) = Ψ′(w)w = µΦ′(w)w = µ(p + 1)Φ(w) = µ(p + 1).

Consequently, µ > 0 and we deduce from (6.4) that u := µ1/(p−1)w is a nonnegativevariational solution of (6.1), u ≡ 0. Corollary 3.4 guarantees that u is a classicalsolution and the strong maximum principle shows u > 0 in Ω.

Remarks 6.3. (i) Annulus. Assume that Ω = x ∈ Rn : 1 < |x| < 2, λ < λ1

and let X denote the space all of radial functions in W 1,20 (Ω). It is easily seen

that X is compactly embedded into the space Y of all radial functions in Lp+1(Ω)for any p > 1 (in fact, X and Y are isomorphic to W 1,2

0

((1, 2)

)and Lp+1

((1, 2)

),

respectively). Moreover, any critical point of E in X is obviously a classical solutionof (6.1). Hence the proof of Theorem 6.2 guarantees the existence of a positiveclassical solution of (6.1) for all p > 1.

(ii) Nonexistence for λ ≥ λ1. If Ω is bounded, λ ≥ λ1 and p > 1 is arbitrary,then (6.1) does not have positive stationary solutions. To see this, it is sufficientto multiply the equation in (6.1) by the first eigenfunction ϕ1 to obtain

0 =∫

Ω

|u|p−1uϕ1 dx + (λ− λ1)∫

Ω

uϕ1 dx > 0

provided u is a positive solution.

Remark 6.4. Unbounded domains. Let Ω = Rn, 1 < p < pS and λ < 0

(notice that 0 is the minimum of the spectrum of −∆ in W 1,2(Rn)). Let X andY denote the space of radial functions in W 1,2(Rn) and Lp+1(Rn), respectively. Ifn ≥ 2, then X is compactly embedded in Y (see [76, Theorem A.I’]) so that wemay use the approach above in order to get a positive solution of (6.1). Moreover,using Schwarz symmetrization it is easy to see that the minimizer of Ψ(u) =12

∫Ω

(|∇u|2 − λu2

)dx in MX := u ∈ X : 1

p+1

∫Ω|u|p+1 dx = 1 is also a minimizer

in the larger set M := u ∈W 1,2(Rn) : 1p+1

∫Ω|u|p+1 dx = 1.

In the case Ω = Rn one can use a similar approach to that used in Theorem 6.2

for functions f = f(u) (or f = f(|x|, u)) which need not be homogeneous. In fact,if one is able to find a minimizer u of 1

2

∫Ω|∇u|2 dx in the set N := u ∈ X :∫

Ω F (u) dx = 1, then there exists σ > 0 such that the function uσ(x) := u(x/σ)solves (6.1). This idea was used in [76], for example. For more recent results onexistence and uniqueness of positive solutions of this problem with f = f(u) werefer to [235], [420] and the references therein.

If f depends on x (and not only on |x|) or if Ω is unbounded and not symmetric,then the situation is more delicate. In some cases, one can use the concentrationcompactness arguments in order to get a solution (see [50] and the referencestherein).

Let us now turn to the critical case p = pS. In view of Corollary 5.2 and theproof of Theorem 6.2, the functional Ψ cannot attain its infimum over the set Mif Ω is starshaped and λ = 0. In other words, denoting

Sλ(u, Ω) :=

∫Ω

[|∇u|2 − λ|u|2

]dx

‖u‖22∗,

Sλ(Ω) := infSλ(u, Ω) : u ∈ W 1,20 (Ω), u = 0

= infSλ(u, Ω) : u ∈ W 1,20 (Ω), ‖u‖2∗ = 1,

the value S0(Ω) cannot be attained if Ω is starshaped. The following propositionshows that the same is true for any Ω = R

n. In particular, this means that thesolution from Remark 6.3(i) (for p = pS and λ = 0) is not a minimizer of S0(·, Ω).

Proposition 6.5. We have S0(Ω1) = S0(Ω2) for any open sets Ω1, Ω2 ⊂ Rn. If

Ω = Rn, then S0(Ω) is not attained.

Proof. Let Ω1, Ω2 ⊂ Rn be open. Since S0(Ω) = S0(x + Ω) for any x ∈ R

n, wemay assume 0 ∈ Ω1 ∩Ω2. Denote wR(x) := w(Rx).

Let ε > 0 and 0 = u1 ∈ W 1,20 (Ω1), S0(u1, Ω1) < S0(Ω1) + ε. Setting u1(x) :=

u(x) if x ∈ Ω1, u1(x) = 0 if x /∈ Ω1, we have u1 ∈ W 1,20 (Rn) = W 1,2(Rn) and

supp (uR1 ) ⊂ Ω2 if R is sufficiently large. Let u2 be the restriction of uR

1 to Ω2.Then u2 ∈W 1,2

0 (Ω2), u2 = 0, and

S0(Ω2) ≤ S0(u2, Ω2) = S0(uR1 , Rn) = S0(u1, R

n)

= S0(u1, Ω1) < S0(Ω1) + ε.

Letting ε → 0 we obtain S0(Ω2) ≤ S0(Ω1). Exchanging the role of Ω1 and Ω2 weobtain the reversed inequality.

Now assume Ω = Rn, u ∈ W 1,2

0 (Ω) and S0(u, Ω) = S0(Ω). We may assumeu ≥ 0, u = 0. Set u(x) := u(x) for x ∈ Ω, u(x) := 0 otherwise. Then S0(u, Rn) =S0(Ω) = S0(Rn), hence u is a minimizer of S0(·, Rn) and the proof of Theorem 6.2shows the existence of µ > 0 such that u is a classical positive solution of theequation −∆u = µ|u|p−1u in R

n. But this is a contradiction with u = 0 outsideΩ.

Remark 6.6. Best constant in Sobolev’s inequality. The function S0(·, Rn)attains its minimum S := S0(Rn) =

(n(n − 2)π

)−1/2(Γ(n)/Γ(n/2))1/n at any

function of the form uε(x− x0), where ε > 0, x0 ∈ Rn and

uε(x) := (ε2 + |x|2)−(n−2)/2.

This was proved by symmetrization techniques in [43] and [508] (for more generalresults of this kind see [111] and the references therein). If we set

Cε :=[n(n− 2)ε2

](n−2)/4,

then the functions Cεuε(·−x0) are the only positive classical solutions of (6.1) withΩ = R

n, p = pS and λ = 0: This follows from Theorems 8.1 and 9.1 below.

Theorem 6.7. Let n ≥ 3 and p = pS. Assume Ω bounded, 0 < λ < λ1. Let S bethe constant from Remark 6.6.

(i) If Sλ(Ω) < S, then there exists u ∈ W 1,20 (Ω) such that u > 0 in Ω and Sλ(Ω) =

Sλ(u, Ω).

(ii) If λ is close to λ1, then Sλ(Ω) < S.

Proof. (i) Let uk be a minimizing sequence for Sλ(·, Ω), ‖uk‖2∗ = 1. Replacinguk by |uk| we may assume uk ≥ 0. Since

(1− λ

λ1

)∫Ω

|∇uk|2 dx ≤∫

Ω

[|∇uk|2 − λu2

k

]dx = Sλ(uk, Ω)→ Sλ(Ω),

the sequence uk is bounded in W 1,20 (Ω) and we may assume uk u in W 1,2

0 (Ω).Due to the embeddings W 1,2

0 (Ω) → L2∗(Ω) and W 1,2

0 (Ω) →→ L2(Ω) we obtainuk u in L2∗

(Ω) and uk → u in L2(Ω). Passing to a subsequence we may assumeuk(x) → u(x) a.e. Given t ∈ [0, 1], denote

ψk = ψk(t) := 2∗(uk + (t− 1)u

)∣∣uk + (t− 1)u∣∣2∗−2

, ψ = ψ(t) := 2∗tu|tu|2∗−2.

Then ψk → ψ a.e. in Ω and ψk, ψ are uniformly bounded in L2∗(Ω), where 2∗ :=(2∗)′ = 2n/(n + 2). Using Vitali’s convergence theorem we obtain∫

Ω

[|uk|2

∗− |uk − u|2

∗]dx =

∫Ω

∫ 1

0

d

dt

∣∣uk + (t− 1)u∣∣2∗

dt dx

=∫ 1

0

∫Ω

ψkudx dt →∫ 1

0

∫Ω

ψudx dt =∫

Ω

|u|2∗dx as k →∞,

hence‖u‖2∗

2∗ = 1− ‖uk − u‖2∗2∗ + o(1),

where o(1) → 0 as k →∞. The weak convergence uk u in W 1,20 (Ω) implies∫

Ω

|∇uk|2 dx =∫

Ω

|∇(uk − u)|2 dx +∫

Ω

|∇u|2 dx + o(1),

hence

Sλ(Ω) = Sλ(uk, Ω) + o(1)

=∫

Ω

|∇(uk − u)|2 dx +∫

Ω

[|∇u|2 − λu2

]dx + o(1)

≥ S‖uk − u‖22∗ + Sλ(Ω)‖u‖22∗ + o(1)

≥ S‖uk − u‖2∗

2∗ + Sλ(Ω)‖u‖2∗

2∗ + o(1)

=(S − Sλ(Ω)

)‖uk − u‖2

∗2∗ + Sλ(Ω) + o(1).

Now S > Sλ(Ω) implies uk → u in L2∗(Ω), hence ‖u‖2∗ = 1. The weak lower

semi-continuity of the norm in W 1,20 (Ω) guarantees

Sλ(u, Ω) ≤ lim infk→∞

Sλ(uk, Ω) = Sλ(Ω),

thus Sλ(u, Ω) = Sλ(Ω). Similarly as in the proof of Theorem 6.2, a suitable positivemultiple of u is a classical positive solution of (6.1) with p = pS , hence u > 0 in Ω.

(ii) Let ϕ1 be the first eigenfunction, ‖ϕ1‖2∗ = 1. Then

Sλ(Ω) ≤ Sλ(ϕ1, Ω) = (λ1 − λ)∫

Ω

ϕ21 dx < S

if λ is close to λ1.

Corollary 6.8. Let n ≥ 3 and p = pS. Assume Ω bounded, 0 < λ < λ1. If λ isclose to λ1, then problem (6.1) has a classical positive solution.

Remarks 6.9. (i) Positive solutions in the critical case [98]. Let Ω bebounded, p = pS ,

λ∗ := infλ ∈ (0, λ1) : Sλ(Ω) < S.

Set uε(x) := (ε + |x|2)−(n−2)/2 (cf. Remark 6.6) and assume 0 ∈ Ω. If n ≥ 4and λ > 0, then careful estimates show Sλ(uεϕ, Ω) < S provided ϕ ∈ D(Ω) isnonnegative, ϕ = 1 in a neighborhood of 0 and ε is small enough. Consequently,λ∗ = 0 in this case and problem (6.1) possesses a positive solution for any λ ∈(0, λ1).

Now let n = 3 and Ω = B1(0). If λ > λ1/4, then Sλ(uεϕ, Ω) < S providedϕ(x) = cos(π|x|/2) and ε is small enough. On the other hand, one can use aPohozaev-type identity for radial functions in order to prove that (6.1) does nothave positive radial solutions if λ ≤ λ1/4. Since any positive solution of (6.1) issymmetric due to [239] we have λ∗ = λ1/4 in this case and the problem possessespositive solutions if and only if λ ∈ (λ1/4, λ1).

Another proof of the above results for Ω = B1(0) based on the ODE techniquescan be found in [40]. The authors use the symmetry of positive solutions u = u(|x|)of (6.1) and the substitution y(t) = u(|x|), t = (n − 2)n−2|x|−(n−2), which trans-forms the problem into the ODE y′′+t−k(λy+ypS ) = 0 with k := 2(n− 1)/(n− 2).

(ii) Uniqueness for p ≤ pS . Uniqueness of positive solutions of (6.1) in thecase Ω = B1(0), p ≤ pS , was established in [239] (if λ = 0), [393] (if λ ≥ 0,p ≤ psg), [310] (if λ < 0, p < pS) and [544], [503] (if λ > 0, p ≤ pS). Some of thesearticles contain also uniqueness results for more general functions f(|x|, u) and forΩ being an annulus.

Uniqueness fails for general bounded domains (see (iii) and (iv) below). On theother hand if Ω satisfies some convexity and symmetry properties, then uniqueness(and non-degeneracy) for positive solutions of (6.1) is true, at least for some valuesof p and/or λ (see [147], [118], [256], for example).

(iii) Nonradial minimizers. Let Ω = x : 1 < |x| < 2, n ≥ 3, λ = 0 andp > 1. Set

S(u, Ω, p) :=

∫Ω |∇u|2 dx

‖u‖2p+1

,

S(Ω, p) := infS(u, Ω, p) : u ∈W 1,20 (Ω) u = 0,

Sr(Ω, p) := infS(u, Ω, p) : u ∈W 1,20 (Ω) u = 0, u is radial.

By Remark 6.3(i), problem (6.1) with λ = 0 has a positive radially symmetricsolution u which minimizes S(·, Ω, p) in the class of radial functions. Since S(Ω, pS)is not attained (see Proposition 6.5), we have S(Ω, pS) < Sr(Ω, pS). It is easy tosee that the functions p → S(Ω, p) and p → Sr(Ω, p) are continuous. Consequently,S(Ω, p) < Sr(Ω, p) also for p < pS , p close to pS . Since S(Ω, p) is attained in thesubcritical case, the corresponding (positive) minimizer is not radially symmetric.

(iv) Effect of the topology of domain. Let Ω be bounded, n ≥ 3, p = pS

and λ = 0. The above considerations show that (6.1) has a positive solution if Ωis an annulus but it does not possess positive solutions if Ω is starshaped. It wasproved in [47] that this problem has positive solutions whenever the homology ofdimension d of Ω with Z2 coefficients is nontrivial for some positive integer d. Inparticular, this is true when n = 3 and Ω is not contractible. On the other hand,there are several examples showing that positive solutions do exist even if Ω iscontractible (see [149], for example).

Let Ω be bounded and let its Ljusternik-Schnirelman category be bigger than1. If p < pS , then problem (6.1) admits multiple positive solutions whenever p isclose to pS or λ < 0 and |λ| is large enough (see [68]); the same is true if p = pS ,λ > 0 is small and n ≥ 4 (see [456], [323]). Again, this topological condition onΩ is not necessary (see [148], where multiple positive solutions are constructed forany p < pS, λ = 0 and Ω being starshaped, and see [408] for the critical case).

(v) Critical case in the unit ball. Let Ω = B1(0), n ≥ 3, p = pS and considerradial (classical) solutions of (6.1).

Due to Corollary 5.2, nontrivial solutions do not exist if λ ≤ 0. Denote by X thespace of all radial functions in W 1,2

0 (Ω) and let λrk denote the k-th eigenvalue of −∆

in X (λrk = k2π2 if n = 3). The corresponding radial eigenfunction ϕr

k (consideredas a function of r := |x|) has (k−1) zeros in (0, 1) and each point (0, λr

k) ∈ X×R isa bifurcation point for (6.1) (see [451]). The corresponding bifurcation branch Bk

of nontrivial solutions is an unbounded continuous curve and u has (k − 1) zerosfor any (u, λ) ∈ Bk. Moreover, there exists µk := limλ : (u, λ) ∈ Bk, ‖u‖X →∞,k = 1, 2, . . . , and we have µk = (k − 1

2 )2π2 if n = 3, µ1 = 0 if n ≥ 4, µk+1 = λrk if

n = 4, 5, µk+1 ∈ (0, λrk) if n = 6, µk = 0 if n ≥ 7 (see Figure 1 and [40], [41], [42],

[39]).Denote µk := infλ : (u, λ) ∈ Bk. The results mentioned in (i) and (ii) imply

µ1 = µ1 = λ1/4 if n = 3, µ1 = µ1 = 0 if n ≥ 4. Similarly, [34] and [234] implyµ2 = µ2 if n = 4, µ2 < µ2 if n = 5 but the relation between µ2 and µ2 forn ∈ 3, 6 seems to be an open problem.

Denote also λ∗ := infµk : k ≥ 2. Then λ∗ > 0 provided n ≤ 6 (see [39]).On the other hand, problem (6.1) with Ω = B1(0), n ≥ 4, p = pS and λ > 0has infinitely many nontrivial solutions in W 1,2

0 (Ω) (see [212]). Consequently, ifn ∈ 4, 5, 6 and λ < λ∗, then all these solutions (except for ±u1 where u1 denotesthe unique positive solution) have to be nonradial. The existence of (nonradialsign-changing) solutions for Ω = B1(0), n = 3 and λ ∈ (0, λ1/4] seems to be open.

Many interesting results on singular radial solutions of (6.1) for Ω = B1(0) andp > 1 can be found in [73].

Remarks 6.10. Supercritical case. Let n ≥ 3, p > pS .(i) If λ = 0, then the analogue of the result of [47] mentioned in Remark 6.9(iv)

does not hold (see [406], [407]).

‖u‖

0 λ1µ1 λr2µ2

n = 3

B1 B2

µ1 = 0 λ1 = µ2 λr2

n = 5

µ1 = 0 λ1 = µ2 λr2

n = 4

µ1 = 0 λ1µ2 λr2

n = 6

µk = 0 λ1 λr2

n ≥ 7

Figure 1: Bifurcation diagrams for radial solutions of (6.1)with p = pS and Ω = B1(0).

(ii) Let Ω = B1(0). Then the points (0, λrk) from Remark 6.9(v) are bifurcation

points for (6.1) also in this case. Let Bk(p) denote the corresponding bifurca-tion branch and let µk(p), µk(p) have similar meaning as in Remark 6.9(v). Ifn > 6, assume also p < pZZ :=

(n + 1 −

√2n− 3

)/(n − 3 −

√2n− 3

). Then

‖u‖

λ10p < pS

0 λ1λ1/4p = pS , n = 3

0 λ1

p = pS , n > 3

0 λ1

p > pS , n ≤ 6

Figure 2: Bifurcation diagrams for positive solutionsof (6.1) with Ω = B1(0).

0 < µ1(p) < µ1(p) < λ1 and problem (6.1) has infinitely many radial positive(classical) solutions if λ = µ1(p) (see Figure 2 and [546], [104], [365]). It is notclear whether the condition p < pZZ for n > 6 is optimal, but some restrictions onn or p for this behavior of B1(p) may be expected. In fact, bifurcation diagramsfor positive solutions of the related problem

−∆u = λ(1 + u)p, x ∈ B1(0),

u = 0, x ∈ ∂B1(0),

(6.5)

in the supercritical case are completely different for p < pJL and p ≥ pJL, wherepJL is defined in (9.3) (see Figure 3 and [293]).

Note also that the same diagrams as in Figure 3 are true for the problem

−∆u = λeu, x ∈ B1(0),

u = 0, x ∈ ∂B1(0),

(6.6)

and the three cases I, II and III correspond to n ≤ 2, 3 ≤ n ≤ 9 and n ≥ 10,respectively.

7. Minimax methods 29

‖u‖

0 λ

I. p ≤ pS

0 λ

II. pS < p < pJL

0 λ

III. p ≥ pJL

Figure 3: Bifurcation diagrams for positive solutions of (6.5).

7. Minimax methods

In this section we look for saddle points of the energy functional E defined in(3.5) by minimax methods. Throughout this section we assume that f satisfies thegrowth assumption (3.4) so that E is a C1-functional in the Hilbert space W 1,2

0 (Ω)and its critical points correspond to (variational) solutions of (2.1).

Even if we considered a finite-dimensional space X = R2 and a smooth func-

tional E : X → R, then (looking at the graph of E as the earth’s surface) exis-tence of a saddle (mountain pass) on a mountain range between two valleys is notclear, in general. For example, if E : R

2 → R : (x, y) → ex − y2, A0 = (0,−2),A1 = (0, 2), then any path from A1 to A2 in R

2 has to cross the line y = 0 whereE > 0 > maxE(A1), E(A2), but the functional E does not possess critical pointsat all. If one looks for a point with a minimal height on the “mountain range” de-scribed by the graph of E on (x, y) : y = 0, then any minimizing sequence hasthe form (xk, 0), where xk → −∞. In particular, it is not compact and we cannotchoose a subsequence converging to the desired saddle point. Therefore, dealingwith abstract functionals E in a real Banach space X , we shall need additionalinformation on E which will prevent the problem mentioned above.

Definition 7.1. A sequence uk in X is called a Palais-Smale sequence if thesequence E(uk) is bounded and E′(uk) → 0. We say that E satisfies condition(PS) if any Palais-Smale sequence is relatively compact. We say that E satisfiescondition (PS)β (Palais-Smale condition at level β) if any sequence uk satisfyingE(uk)→ β, E′(uk)→ 0, is relatively compact. A real number β is called a criticalvalue of E if there exists u ∈ X with E′(u) = 0 and E(u) = β.

The following mountain pass theorem is due to [23]. Our proofs of this theoremand Theorems 7.4, 7.8 below closely follow those in [505, Chapter II].

Theorem 7.2. Suppose that E ∈ C1(X) satisfies (PS). Let u0, u1 ∈ X,

M := maxE(u0), E(u1),P := p ∈ C([0, 1], X) : p(0) = u0, p(1) = u1,β := inf

p∈Pmax

t∈[0,1]E(p(t)).

(7.1)

If β > M , then β is a critical value of E.

Given β ∈ R and δ > 0, denote

Nδ = Nδ(β) := u ∈ X : |E(u)− β| ≤ δ, ‖E′(u)‖ ≤ δ

and Eβ := u ∈ X : E(u) < β.In the proof of Theorem 7.2 we shall need the following deformation lemma.

Lemma 7.3. Suppose that E ∈ C1(X) and let Nδ(β) = ∅ for some δ < 1. Chooseε = δ2/2. Then there exists a continuous mapping Φ : X × [0, 1]→ X such that(i) Φ(u, t) = u whenever t = 0 or |E(u)− β| ≥ 2ε,(ii) t → E(Φ(u, t)) is nonincreasing for all u,(iii) Φ(Eβ+ε, 1) ⊂ Eβ−ε.In addition, Φ(·, t) is odd if E is even.

Proof. In order to avoid all technicalities we shall assume, in addition, that E ∈C2(X) and X is a Hilbert space. Notice that these assumptions are satisfied in ourapplications if f has the form (2.2), for example (and see e.g. [505] for the proofin the general case).

Choose functions ϕ, ψ : R → [0, 1] such that ϕ is smooth, ϕ(t) = 1 for |t−β| ≤ ε,ϕ(t) = 0 for |t− β| ≥ 2ε, ψ(t) = 1 for t ≤ 1 and ψ(t) = 1/t for t > 1. The vectorfield

F : X → X : u → −ϕ(E(u)

)ψ(‖E′(u)‖

)∇E(u)

is bounded and locally Lipschitz. Consequently, the initial value problem

Φt(u, t) = F(Φ(u, t)

), for t ∈ [0, 1],

Φ(u, 0) = u

has a unique solution for any u ∈ X . The function Φ defined in this way is obviouslycontinuous and satisfies (i). Denoting v := Φ(u, t) we have

d

dtE(Φ(u, t)

)=

d

dtE(v) = E′(v)F(v) = −ϕ

(E(v)

)ψ(‖E′(v)‖

)‖E′(v)‖2 ≤ 0,

thus (ii) is true.

Assertion (iii) will be proved by a contradiction argument. Let u ∈ Eβ+ε andassume Φ(u, 1) /∈ Eβ−ε. Then (ii) implies |E

(Φ(u, t)

)−β| ≤ ε < δ for t ∈ [0, 1],

hence Nδ = ∅ implies ‖E′(Φ(u, t))‖ ≥ δ for t ∈ [0, 1]. Using this estimate and the

properties of the functions ϕ, ψ we get

E(Φ(u, 1)

)= E(u) +

∫ 1

0

d

dtE(Φ(u, t)

)dt

= E(u)−∫ 1

0

ϕ(. . . )︸︷︷︸=1

ψ(. . . )‖E′(Φ(u, t))‖2︸︷︷︸

≥δ2

dt

< β + ε− δ2 ≤ β − ε,

a contradiction.

Proof of Theorem 7.2. Assume that β is not a critical value of E. Then it is easyto use condition (PS) in order to find δ > 0 such that Nδ(β) = ∅. We may assumeδ < 1, δ2 < β −M . Let ε := 1

2δ2 be from Lemma 7.3. By the definition of β thereexists p ∈ P such that maxt∈[0,1] E

(p(t)

)< β+ε. Since E(ui) ≤M < β−δ2 = β−2ε

for i = 0, 1, Lemma 7.3(i) guarantees that p1 : t → Φ(p(t), 1

)is an element of P .

Now Lemma 7.3(iii) implies maxt∈[0,1] E(p1(t)

)≤ β − ε, which contradicts the

definition of β.

The next theorem is again due to [23]. It represents a symmetric variant ofTheorem 7.2 and we will use it for the proof of existence of infinitely many solutionsof problem (2.1).

Theorem 7.4. Suppose that E ∈ C1(X) is even and satisfies (PS). Let X+, X−

be closed subspaces of X with dimX− = codim X+ + 1 <∞. Let E(0) = 0 and letthere exist α, ρ, R > 0 such that E(u) ≥ α for all u ∈ S+

ρ := u ∈ X+ : ‖u‖ = ρand E(u) ≤ 0 for all u ∈ X−, ‖u‖ ≥ R. Set

Γ := h ∈ C(X, X) : h is odd, h(u) = u if E(u) ≤ 0,β := inf

h∈Γmaxu∈X−

E(h(u)

).

Then β is a critical value of E, β ≥ α.

The proof of the above theorem will be almost the same as the proof of Theo-rem 7.2 provided we prove the following Intersection Lemma.

Lemma 7.5. If ρ > 0 and h ∈ Γ, then h(X−) ∩ S+ρ = ∅.

Proof of Theorem 7.4. Lemma 7.5 implies β ≥ α. Assume that β is not acritical value of E. Then Nδ(β) = ∅ for some δ > 0 and we may assume δ < 1,δ2 < α. Let ε := δ2/2 and Φ be from Lemma 7.3 and choose h ∈ Γ such that

E(h(u)

)< β + ε for all u ∈ X−. Set h1(u) := Φ

(h(u), 1

). Then h1 ∈ Γ and

E(h1(u)

)= E

(Φ(h(u), 1

))< β− ε, due to Lemma 7.3(iii). But this contradicts the

definition of β.

In the proof of Lemma 7.5 we shall need the notion of Krasnoselskii genus.

Definition 7.6. Let A be the set of all closed subsets of X satisfying A = −A.If A ∈ A, then we set γ(A) := 0 if A = ∅ and

γ(A) := infm : ∃h ∈ C(A, Rm \ 0), h odd

otherwise.

The following proposition is proved in [505, Propositions II.5.2 and II.5.4]:

Proposition 7.7. Suppose that A, A1, A2 ∈ A and h ∈ C(X, X) is odd. Then thefollowing is true:(1) γ(A) ≥ 0, γ(A) = 0 if and only if A = ∅,(2) if A1 ⊂ A2, then γ(A1) ≤ γ(A2),(3) γ(A1 ∪A2) ≤ γ(A1) + γ(A2),(4) γ(A) ≤ γ

(h(A)

),

(5) if A is compact and 0 /∈ A, then γ(A) < ∞ and there exists a symmetricneighborhood U of A such that U ∈ A and γ(A) = γ(U).(6) Let D be a bounded symmetric neighborhood of zero in Y , where Y is a subspaceof X with m := dim(Y ) < ∞, and let ∂Y D denote the boundary of D in Y . Thenγ(∂Y D) = m.

Proof of Lemma 7.5. Let ρ > 0 and h ∈ Γ. Set R1 := maxR, ρ, B−R1

:=u ∈ X− : ‖u‖ < R1 and Sρ := u ∈ X : ‖u‖ = ρ. Since E(u) ≤ 0 for u ∈X−, ‖u‖ ≥ R, we have ‖h(u)‖ = ‖u‖ > ρ for all u ∈ X−, ‖u‖ > R1, henceh(X−) ∩ Sρ = h

(B−

R1

)∩Sρ is compact. In particular, A := h(X−) ∩ S+

ρ fulfills theassumptions of Proposition 7.7(5), thus there exists its symmetric neighborhoodU with γ(U) = γ(A). By (2) and (3) in Proposition 7.7 we obtain

γ(A) = γ(U) ≥ γ(h(X−) ∩ Sρ ∩ U

)≥ γ

(h(X−) ∩ Sρ

)−γ(B), (7.2)

where Sρ := u ∈ X : ‖u‖ = ρ and B := h(X−) ∩ Sρ \ U . Let Z be a directcomplement of X+ in X and let π : X → Z denote the projection along X+. SinceU is a neighborhood of h(X−)∩ S+

ρ , we get B ∩X+ = ∅, hence 0 /∈ π(B) and thedefinition of γ implies

γ(B) ≤ dimZ = codim X+. (7.3)

Now (2) and (4) in Proposition 7.7 guarantee γ(h(X−)∩Sρ) ≥ γ

(h−1(Sρ)∩X−).

Since h(0) = 0 and h(u) = u for u ∈ X−, ‖u‖ > R, the set h−1(Sρ)∩X− containsthe relative boundary of u ∈ X− : ‖h(u)‖ < ρ which is a symmetric bounded

neighborhood of zero in X−. Consequently, using (2) and (6) in Proposition 7.7we arrive at

γ(h(X−) ∩ Sρ

)≥ dimX− = codim X+ + 1. (7.4)

Now (7.2)–(7.4) imply γ(A) ≥ 1, hence A = ∅.

Theorems 7.2 and 7.4 guarantee the following solvability result.

Theorem 7.8. Assume Ω bounded. Let f be a Caratheodory function, and letthere exist p < pS, R > 0 and µ > 2 such that |f(x, u)| ≤ C(1 + |u|p) for allx ∈ Ω, u ∈ R and f(x, u)u ≥ µF (x, u) > 0 for all x ∈ Ω and |u| > R.(i) If there exist c < λ1 and ρ ∈ (0, 1) such that f(x, u)/u ≤ c for all x ∈ Ω and|u| < ρ, then there exists a positive solution of (2.1).(ii) If f(x,−u) = −f(x, u) for all x ∈ Ω and u ∈ R, then there exists a sequenceuk of solutions of (2.1) with E(uk)→∞ as k →∞.

Proof. The energy functional E associated with (2.1) is C1. Let us first verify thatE satisfies condition (PS). Let uk be a Palais-Smale sequence. Denote |u|1,2 :=(∫

Ω |∇u|2 dx)1/2 and notice that this is an equivalent norm in X := W 1,2

0 (Ω). Then

o(1 + |uk|1,2) = −E′(uk)uk = −|uk|21,2 +∫

Ω

f(x, uk)uk dx

=(µ

2− 1)|uk|21,2 +

∫Ω

[f(x, uk)uk − µF (x, uk)

]dx− µE(uk)

≥(µ

2− 1)|uk|21,2 − C1,

where C1 > 0 is independent of k. Consequently, the sequence uk is bounded inX . We have ∇E(u) = u +F1(u), where F1 is compact.2 Since uk is bounded inX , we may assume (passing to a subsequence if necessary) F1(uk) → w in X forsome w ∈ X . Since o(1) = ∇E(uk) = uk −F1(uk), we obtain uk → w, hence ukis relatively compact.

(i) We will use Theorem 7.2. In order to get a positive solution, let us definef(x, u) := f(x, u) if u ≥ 0, f(x, u) = 0 otherwise, F (x, u) :=

∫ u

0 f(x, s) ds, E(u) :=12

∫Ω|∇u|2 dx−

∫Ω

F (x, u) dx, and notice that E is C1 and satisfies condition (PS).Set u0 := 0, then E(u0) = 0. The assumption f(x, u)/u ≤ c for |u| < ρ guarantees|F (x, u)| ≤ (c/2)u2 for |u| < ρ. If |u| ≥ ρ, then the growth assumption |f(x, u)| ≤C(1 + |u|p) implies

|F (x, u)| ≤ C(|u|+ |u|p+1) ≤ (c/2)u2 + C2|u|p+1,

2The Nemytskii mapping F : Lp+1(Ω) → L(p+1)′ (Ω) : u → f(·, u) is continuous. The embed-

ding Ip : X → Lp+1(Ω) is compact, hence the dual mapping I′p :(Lp+1(Ω)

)′→ X′ is compact

as well. Let R : X′ → X denote the Riesz isomorphism in the Hilbert space X (thus RE′(u) =

∇E(u)) and let J : L(p+1)′ (Ω) → (Lp+1(Ω)

)′be the isomorphism defined by (Jw)u =

∫Ω uw dx

for u ∈ Lp+1(Ω). Then ∇E(u) = u + F1(u), where F1 : X → X : u → RI′pJFIp(u) is compact.

where C2 := C(1 + ρ−p). Consequently, if Cp denotes the norm of the embeddingX → Lp+1(Ω), then

E(u) ≥ 12

∫Ω

|∇u|2 dx− c

2

∫Ω

u2 dx− C2

∫Ω

|u|p+1 dx

≥(1

2− c

2λ1− C2C

p+1p |u|p−1

1,2

)|u|21,2 ≥ α > 0

provided |u|1,2 = δ is small enough. Now the assumption f(x, u)u ≥ µF (x, u) > 0implies d

du

(u−µF (x, u)

)≥ 0 for u > R, hence F (x, u) ≥ b(x)uµ for u > R, where

b(x) := R−µF (x, R) > 0. Hence, fixing u ∈ X , u > 0 in Ω, denoting

A(u) :=12

∫Ω

|∇u|2 dx, B(u) :=∫

Ω

b(x)uµ dx > 0, (7.5)

and taking t > 0, we obtain

E(tu) = E(tu) ≤ t2A(u)− tµB(u) + C3 → −∞ as t→∞,

where we used the estimate∫0<tu≤R

[b(x)(tu)µ − F (x, tu)

]dx ≤ C3

with C3 independent of t and u. Hence, choosing u1 := tu with t large enoughwe have E(u1) < 0. Let β be the number defined in Theorem 7.2. Since anypath joining u0 and u1 has to intersect the sphere Sδ := u : |u|1,2 = δ, wehave β ≥ α > 0 and Theorem 7.2 guarantees the existence of a solution u withE(u) ≥ α. Since f(x, u) = 0 for u ≤ 0, the maximum principle implies u ≥ 0. NowE(u) = E(u) > 0, hence u = 0 and using the maximum principle again we obtainu > 0 in Ω.

(ii) Choose a positive integer k. Let X− denote the linear hull of ϕ1, ϕ2, . . . , ϕk,and X+ be the closure of the linear hull of ϕk, ϕk+1, . . . . The growth conditionon f guarantees |F (x, u)| ≤ CF (1 + |u|p+1) for suitable CF > 0. Set q := pS ifn ≥ 3 and choose q > p otherwise. Let C4 := CF Cp+1−r

q and C5 := CF |Ω|, whereCq denotes the norm of the embedding Iq : W 1,2

0 (Ω) → Lq+1(Ω), r ∈ (0, p + 1) isdefined by r/2 + (p + 1 − r)/q = 1 and |Ω| denotes the measure of Ω. If u ∈ X+

and ‖u‖ = ρ := ρk :=(λ

r/2k /(4C4)

)1/(p−1), then

E(u) ≥ 12

∫Ω

|∇u|2 dx− CF

∫Ω

|u|p+1 dx− C5

≥ 12|u|21,2 − CF ‖u‖r

2‖u‖p+1−rq+1 − C5

≥(1

2− C4λ

−r/2k |u|p−1

1,2

)|u|21,2 − C5

=(1

2− C4λ

−r/2k ρp−1

)ρ2 − C5 = C6λ

r/(p−1)k − C5,

where C6 = (4pC4)−1/(p−1). Denote α = αk := infE(u) : u ∈ X+, |u|1,2 = ρ.Since λk →∞ as k →∞, we have αk →∞.

On the other hand, estimates in (i) show E(tu) ≤ t2A(u)− tµB(u)−C3, whereA, B are defined in (7.5). Since A(u) = 1/2 for |u|1,2 = 1 and C7 := infB(u) :u ∈ X−, |u|1,2 = 1 > 0, we have

E(u) ≤ 12|u|21,2 − C7|u|µ1,2 + C3 for all u ∈ X−,

hence the assumptions of Theorem 7.4 are satisfied for any k large enough and weobtain a sequence of critical points uk of E satisfying E(uk) ≥ αk. (In fact, a morecareful choice of ρ above enables one to use Theorem 7.4 for any k.)

Remarks 7.9. (i) Linking. Let f be differentiable in u, f(x, 0) = 0, f(x, u)/u ≥fu(x, 0) for all x ∈ Ω and u ∈ R. If the assumption f(x, u)/u ≤ c < λ1 for usmall in Theorem 7.8(i) fails, then one can use a modification of the mountainpass theorem, so called “linking”, in order to prove the existence of a nontrivialsolution of (2.1) (see [505, Section II.8] and the references therein).

(ii) Perturbation results. Consider the problem

−∆u = |u|p−1u + ϕ, x ∈ Ω,

u = 0, x ∈ ∂Ω,

(7.6)

where Ω ⊂ Rn is bounded, 1 < p < pS and ϕ ∈ W−1,2(Ω) :=

(W 1,2

0 (Ω))′.

Theorem 7.8(ii) guarantees existence of infinitely many solutions of (7.6) providedϕ = 0. The same result is known to be true for ϕ belonging to a residual set inW−1,2(Ω) (see [45]) and for all ϕ ∈ W−1,2(Ω) provided p(n − 2) < n (see [300,Theoreme V.4.6.]; see also [506], [46], [452] and [48]). On the other hand, if n > 2,p ∈ [psg, pS) and ϕ is a general (smooth) function, then even the solvability of(7.6) seems to be open.

(iii) Unbounded domains. If Ω = Rn, then the existence of infinitely many

solutions of (2.1) is known in many cases as well. We refer to [76], [140], [139], [9]and the references therein.

(iv) Critical case. Let Ω ⊂ Rn be bounded, p = pS and λ > 0. If n ≥ 7,

then problem (6.1) possesses infinitely many solutions, see [162]. Such a result isknown for any n ≥ 4 if the domain Ω exhibits suitable symmetries (see [212])but not for general domains (cf. also the results for Ω being a ball mentioned inRemark 6.9(v)). If n = 6 and λ ∈ (0, λ1), then (6.1) has at least two (pairs of)solutions for any bounded Ω, see [117]. Recall also that if λ ≤ 0, p ≥ pS andΩ is starshaped, then (6.1) does not possess nontrivial classical solutions due toCorollary 5.2.

8. Liouville-type results

In order to prove a priori bounds for positive solutions of (2.1) with f(x, u) ∼ up

as u → +∞, 1 < p < pS (see the rescaling method in Section 12), it will beimportant to know that the problems

−∆u = up, x ∈ Rn (8.1)

and−∆u = up, x ∈ R

n+,

u = 0, x ∈ ∂Rn+,

(8.2)

do not possess positive bounded (classical) solutions. Here Rn+ denotes the half-

space x ∈ Rn : xn > 0. In fact, we shall see in Chapter II that these Liouville-

type results have important applications for parabolic problems as well. In thissection we even prove that these problems do not possess any positive classical so-lution. The following two results are due to [240], [241], except for Theorem 8.1(ii)which was proved in [108].

Theorem 8.1. Let Ω = Rn and p > 1.

(i) If p < pS, then equation (8.1) does not possess any positive classical solution.

(ii) If p = pS, then any positive classical solution of (8.1) is radially symmetricwith respect to some point.

Theorem 8.2. Let 1 < p ≤ pS. Then problem (8.2) does not possess any positiveclassical solution.

We will see in the next section that the condition p < pS is optimal for nonex-istence in R

n. However, in the case of a half-space and if we consider only boundedpositive solutions, nonexistence is known for a larger range of exponents, namelyp < p′S := (n + 1)/(n − 3)+ (note that p′S is the Sobolev exponent in (n − 1)dimensions). This result is due to [150].

Theorem 8.3. Let 1 3.

Then problem (8.2) does not possess any positive, bounded classical solution.

On the other hand, under a stronger assumption on p, one can extend thenonexistence result in R

n to elliptic inequalities. The following result is due to[238].

8. Liouville-type results 37

Theorem 8.4. Let 1 < p ≤ psg. Then the inequality

−∆u ≥ up, x ∈ Rn (8.3)

does not possess any positive classical solution.

Remarks 8.5. (i) It seems unknown if the condition p ≤ pS is optimal for thenonexistence of positive solutions of (8.2). In the case of positive bounded solutions,the results recently announced in [178] indicate that the condition p < p′S can beimproved, but the optimal exponent seems to remain unknown.

(ii) The condition p ≤ psg in Theorem 8.4 is optimal, as shown by the explicitexample u(x) = k(1 + |x|2)−1/(p−1) with n ≥ 3, p > psg and k > 0 small enough.

(iii) Consider the inequality −∆u ≥ up in the half-space Rn+ (no boundary

conditions required). Then nonexistence of positive solutions holds whenever p ≤(n + 1)/(n− 1) (see [74]).

(iv) Consider “quasi-solutions” of (8.1), i.e. (nonnegative) functions satisfyingthe double inequality

aup ≤ −∆u ≤ up, x ∈ Rn, (8.4)

for some a ∈ (0, 1). It is shown in [509] that if 1 psg and a ∈ (0, 1) is small enough,then (8.4) possesses positive solutions u ∈ C2(Rn). Note that a simple example isprovided by the function u(x) = k(1 + |x|2)−1/(p−1) with k > 0 large enough.

We start by proving Theorem 8.4, which is much easier than Theorems 8.1 and8.2. The following proof (cf. [74], [501], [372]) is based on a rescaled test-functionargument, and it is different and simpler than the original proof of [238].

Proof of Theorem 8.4. Take ξ ∈ D(B1), 0 ≤ ξ ≤ 1, with ξ = 1 for |x| ≤ 1/2,and let m = 2p/(p− 1). Fix R > 0 and define ϕR(x) = ξm(x/R). We observe that

∆ϕR = mR−2[ξm−1∆ξ + (m− 1)ξm−2|∇ξ|2

](x/R)

hence|∆ϕR| ≤ CR−2ξm−2(x/R)χ|x|>R/2 = CR−2ϕ

1/pR χ|x|>R/2.

Multiplying (8.3) by ϕR, integrating by parts, and using Holder’s inequality, weobtain∫

Rn

upϕR ≤ −∫

Rn

u ∆ϕR ≤ CR−2

∫R/2<|x|<R

u ϕ1/pR

≤ CRn(p−1)/p−2(∫

R/2<|x|<R

upϕR

)1/p

.

(8.5)

In particular, it follows that∫upϕR ≤ CRn−2p/(p−1). (8.6)

If p < psg, i.e. n− 2p/(p− 1) < 0, then this implies u ≡ 0 upon letting R →∞. Ifp = psg, then (8.6) implies

∫Rn up < ∞. Therefore, the RHS of (8.5) goes to 0 as

R →∞ and we again conclude that u ≡ 0.

Theorem 8.1 is much more delicate. Note that, in view of Theorem 8.4, we mayrestrict ourselves to n ≥ 3. We will give a first proof of Theorem 8.1(i) which,like the original proof of [240], is based on integral estimates for (local) positivesolutions (cf. Proposition 8.6 below). Here we essentially follow the (simplified)treatment of [83]. Next, we will prove Theorem 8.2 by using moving plane argu-ments, following [241]. We will then give a second, completely different proof ofTheorem 8.1(i), also based on moving planes arguments, which is due to [123], [78]and allows us to prove Theorem 8.1(ii) at the same time. We point out that thetechniques of both proofs of Theorem 8.1(i) are important and can be extendedto some other problems (see e.g. Section 21 and [123], [78], respectively). Finally,we will prove Theorem 8.3 following [150]. Note that, although the proofs of bothTheorems 8.2 and 8.3 are based on moving planes arguments, they use differ-ent ideas: reduction to the one-dimensional problem on a half-line for the former,monotonicity and reduction to the (n−1)-dimensional problem in the whole spacefor the latter.

Proposition 8.6. Let 1 max(n(p− 1)/2, p) such that if 0 < u ∈ C2(B1) is a solution of

−∆u = up (8.7)

in B1, then ∫|x|<1/2

ur ≤ C(n, p). (8.8)

Let us assume for the moment that Proposition 8.6 is proved and deduce someconsequences of it. To prove Theorem 8.1(i) it suffices to apply a simple homo-geneity argument.

Proof of Theorem 8.1(i). Assume that u is a positive solution of (8.1). Then,for each R > 0, v(x) := R2/(p−1)u(Rx) solves (8.7) in B1. It follows from Propo-sition 8.6 that∫

|y|<R/2

ur(y) dy = Rn

∫|x|<1/2

ur(Rx) dx

= Rn−2r/(p−1)

∫|x|<1/2

vr(x) dx ≤ C(n, p)Rn−2r/(p−1).

By letting R →∞, we conclude that∫

Rn ur(y) dy = 0, a contradiction.

As another important consequence of Proposition 8.6, we have the followingresult (cf. [151]) concerning singularities of local solutions to (8.7) in arbitrarydomains. Note that when psg 0 such that if 0 max(n(p− 1)/2, p)

be given by Proposition 8.6. We may fix ρ > 1 such that

p− 1ρ

<2r

n. (8.11)

Assume that v is a solution of

−∆v = vp in B := x ∈ Rn : |x| < 1. (8.12)

Let i be a nonnegative integer and assume that, for all ω ⊂⊂ B, there exists aconstant Ci(n, p, ω) > 0 (independent of v) such that

‖v‖Lrρi (ω) ≤ Ci(n, p, ω). (8.13)

Note that (8.13) is true for i = 0 by Proposition 8.6. Since rρi/p > 1 and

p

rρi− 1

rρi+1=

1rρi

(p− 1

ρ

)<

2n

due to (8.12), we may apply Proposition 47.6(ii) to deduce that (8.13) is true withi replaced by i+1. After a finite number of steps, we obtain ‖v‖Lk(ω) ≤ C(n, p, ω)for some k > n/2. We may then apply Proposition 47.6(ii) once more to deducethat

v(0) ≤ C(n, p). (8.14)

Now assume that u is a solution of (8.9), fix x0 ∈ Ω and let R := dist(x0, ∂Ω).Then v(x) := R2/(p−1)u(x0 + Rx) solves (8.12) and the conclusion follows from(8.14).

Remarks 8.8. (i) More general nonlinearities. Results similar to Theorem 8.7for more general nonlinearities can be found in [240], [83], [473], [424]. In particular,universal singularity estimates of the type of (8.10) are established in [424] whenthe nonlinearity up is replaced by any f(x, u) such that f(x, u) ∼ up, as u → ∞,with 1 < p < pS . The method of proof is different: The estimate is directly deducedfrom the Liouville-type Theorem 8.1(i) by using rescaling and doubling arguments(see Theorem 26.8 and Lemma 26.11 below for a similar approach in the paraboliccase).

(ii) Singularities of quasi-solutions. For “quasi-solutions” of (8.1) (cf. Re-mark 8.5(iv)), the local behavior near an isolated singularity was studied in [509].Let Ω = B(0, 1) \ 0. If psg psg and a ∈ (0, 1)is small enough, then there exist solutions of (8.15) with arbitrarily large growthrates as x→ 0.

On the other hand, by a straightforward modification of the proof of [424,Theorem 2.1], one can show the following uniform and global property: For eachp ∈ (1, pS), there exist a = a(n, p) ∈ (0, 1) and C(n, p) > 0 such that, for anydomain Ω ⊂ R

n, estimate (8.10) is true for any positive solution u ∈ C2(Ω) of(8.15). Note that, as a consequence of this estimate, one recovers the nonexistenceresult in Remark 8.5(iv).

(iii) Radial supercritical case. When p ≥ pS , Ω = BR and u is a radialpositive classical solution of (8.9), a similar argument as in Remark 4.3(iii) showsthat u(r) ≤ C(R−r)−2/(p−1), 0 ≤ r < R, for some C > 0. However the constant Ccannot depend only on n, p, since otherwise this would imply nonexistence of radialpositive classical solutions of (8.9) for Ω = R

n and p ≥ pS , hence contradictingTheorem 9.1 below.

We now turn to the proof of Proposition 8.6. It is based on a key gradientestimate for local solutions of (8.7) (see (8.22) below). To establish this estimate,we prepare the following lemma, which provides a family of integral estimatesrelating any C2-function with its gradient and its Laplacian. The proof relies onthe Bochner identity (8.18), on the change of variable v = uk+1, and on test-functions of the form ϕvm.

In the rest of this section, we use the notation∫

=∫Ω for simplicity.

Lemma 8.9. Let Ω be an arbitrary domain in Rn, 0 ≤ ϕ ∈ D(Ω), and 0 < u ∈

C2(Ω). Fix q ∈ R and denote

I =∫

ϕuq−2|∇u|4, J =∫

ϕuq−1|∇u|2∆u, K =∫

ϕuq(∆u)2.


Then, for any k ∈ R with k = −1, there holds

αI + βJ + γK ≤ 12

∫uq|∇u|2∆ϕ+

∫uq[∆u + (q− k)u−1|∇u|2

]∇u · ∇ϕ, (8.16)

where

α = −n− 1n

k2 + (q − 1)k − q(q − 1)2

, β =n + 2

nk − 3q

2, γ = −n− 1

n.

Proof. Step 1. We first claim that for all v ∈ C2(Ω), v > 0 and any m ∈ R, thereholds

m(1−m)2

∫ϕvm−2|∇v|4 − 3m

2

∫ϕvm−1|∇v|2∆v − n− 1

n

∫ϕvm(∆v)2

≤ 12

∫vm|∇v|2∆ϕ +

∫ [vm∆v + mvm−1|∇v|2

]∇v · ∇ϕ.

(8.17)First note that, by density, it suffices to prove (8.17) for v ∈ C3(Ω). To prove

the claim, we start from the identity

12∆|∇v|2 = ∇(∆v) · ∇v + |D2v|2, (8.18)

where |D2u|2 =∑

1≤i,j≤n

(uxixj )2. Multiplying by ϕvm and integrating over Ω, we

obtain

T1 + T2 :=∫

ϕvm∇(∆v) · ∇v +∫

ϕvm|D2v|2 =12

∫ϕvm∆|∇v|2 =: T3. (8.19)

Integrating by parts and using ϕ ∈ D(Ω), we can rewrite the first and third termsas follows:

T1 = −∫

(∆v)∇ · (ϕvm∇v)

= −∫

vm(∆v)∇v · ∇ϕ−m

∫ϕvm−1|∇v|2∆v −

∫ϕvm(∆v)2

and

T3 =∫|∇v|2

[12vm∆ϕ + mvm−1∇v · ∇ϕ +

m

2ϕ(vm−1∆v + (m− 1)vm−2|∇v|2)

]=

12

∫vm|∇v|2∆ϕ + m

∫vm−1|∇v|2∇v · ∇ϕ

+m

2

∫ϕvm−1|∇v|2∆v +

m(m− 1)2

∫ϕvm−2|∇v|4.


Moving the first term of T1 to the right of (8.19) and the last two terms of T3 tothe left, it follows that

m(1−m)2

∫ϕvm−2|∇v|4 − 3m

2

∫ϕvm−1|∇v|2∆v +

∫ϕvm|D2v|2

=∫

ϕvm(∆v)2 +12

∫vm|∇v|2∆ϕ +

∫ [vm∆v + mvm−1|∇v|2

]∇v · ∇ϕ.

(8.20)By Cauchy-Schwarz’ inequality (applied with the inner product (A, B) = tr(AB∗)on matrices), we have

(∆v)2 =(tr(D2v)

)2 ≤ tr[(D2v)(D2v)∗

]tr (In) = n|D2v|2. (8.21)

Due to ϕ ≥ 0, Claim (8.17) follows by combining (8.20) and (8.21).Step 2. We set v = uk+1, m = (k + 1)−1(q− 2k), that is q = (k +1)m +2k, and

we compute∫ϕvm−2|∇v|4 = (k + 1)4

∫ϕu(k+1)(m−2)+4k|∇u|4 = (k + 1)4I,

∫ϕvm−1|∇v|2∆v = (k + 1)3

∫ϕu(k+1)(m−1)+3k|∇u|2(∆u + ku−1|∇u|2)

= (k + 1)3(kI + J),∫ϕvm(∆v)2

= (k + 1)2∫

ϕu(k+1)m+2k[(∆u)2 + 2k(∆u)u−1|∇u|2 + k2u−2|∇u|4

]= (k + 1)2(k2I + 2kJ + K),∫

vm(∆v)∇v · ∇ϕ = (k + 1)2∫

u(k+1)m+2k[∆u + ku−1|∇u|2

]∇u · ∇ϕ,

and ∫vm−1|∇v|2∇v · ∇ϕ = (k + 1)3

∫u(k+1)m+2k−1|∇u|2∇u · ∇ϕ.

Substituting in (8.17) and dividing by (k + 1)2, we get[m(1−m)2

(k + 1)2 − 3m

2k(k + 1)− n− 1

nk2]I +

[−3m

2(k + 1)− 2k

n− 1n

]J

− n− 1n

K ≤ 12

∫uq|∇u|2∆ϕ +

∫uq[∆u + (k + m(k + 1))u−1|∇u|2

]∇u · ∇ϕ,

which readily implies the lemma.

Lemma 8.10. (i) Let Ω be an arbitrary domain in Rn, and 0 ≤ ϕ ∈ D(Ω). Let

0 −p, k = −1 anddenote

I =∫

ϕuq−2|∇u|4, K =∫

ϕu2p+q.

Then there holds

αI + δK ≤ 12

∫uq|∇u|2∆ϕ + C

∫ [up+q + uq−1|∇u|2

]|∇u · ∇ϕ|, (8.22)

where C = C(n, p, q, k) > 0 and

α = −n− 1n

k2 + (q− 1)k− q(q − 1)2

, δ =1

p + q

(3q

2− n + 2

nk)− n− 1

n. (8.23)

(ii) Assume that 1 0, 2p + q > n(p− 1)/2. (8.24)

Proof. (i) Since −∆u = up, we have

(p + q)J = −∫

ϕ (p + q)up+q−1|∇u|2 = −∫

ϕ∇u · ∇(up+q)

=∫

ϕ (∆u)up+q +∫

(∇ϕ · ∇u)up+q,

where J is defined in Lemma 8.9, hence

(p + q)J = −∫

ϕu2p+q +∫

(∇ϕ · ∇u)up+q.

Substituting in (8.16), we obtain (8.22).(ii) A simple computation shows that δ > 0 and 2p+q > n(p−1)/2 is equivalent

to

k < k0(q) :=q

2− (n− 1)p

n + 2and q > q0(p) :=

(n− 4)p− n

2.

For k = k0(q), we have

α = α(k0(q)) =n− 1

n

(−q2

4+

(n− 1)pq

n + 2− (n− 1)2p2

(n + 2)2)

+ (q − 1)(q

2− (n− 1)p

n + 2

)− q(q − 1)

2

=n− 1

n

(−q2

4+

(n− 1)pq

n + 2− (n− 1)2p2

(n + 2)2− np(q − 1)

n + 2

)=

n− 1n

(−q2

4− pq

n + 2+

n(n + 2)p− (n− 1)2p2

(n + 2)2).

The discriminant of the above polynomial in q is given by

D =p2 + n(n + 2)p− (n− 1)2p2

(n + 2)2=

np[(n + 2)− (n− 2)p](n + 2)2

> 0.

Therefore we have α(k0(q)) > 0 for q ∈ (− 2pn+2 − 2

√D,− 2p

n+2 + 2√

D). Moreover,− 2p

n+2 > q0(p) is equivalent to n(n + 2) > (n2 − 2n − 4)p, which is true due top < pS . Choosing

q = − 2p

n + 2and k = k0(q)− =

(− np

n + 2

)−(with k = −1),

we see that (8.24) is fulfilled.

Proof of Proposition 8.6. Take q, k as in Lemma 8.10(ii) and Ω = B1. We shallestimate the terms on the RHS of (8.22). Let ξ ∈ D(Ω), be such that ξ = 1 for|x| ≤ 1/2 and 0 ≤ ξ ≤ 1. Put θ = (3p + 1 + 2q)/2(2p + q) ∈ (0, 1). By takingϕ = ξm with m = 2/(1− θ), we have

|∇ϕ| ≤ Cξm−1 ≤ Cϕθ, |∆ϕ| ≤ Cξm−2 ≤ Cϕθ. (8.25)

Fix ε > 0. Using Young’s inequality under the form

xyz ≤ εxa + εyb + C(ε)zc, a−1 + b−1 + c−1 = 1,

and (8.25), we obtain∫uq|∇u|2∆ϕ =

∫ (ϕ1/2u(q−2)/2|∇u|2

)(ϕ(q+2)/2(2p+q)u(q+2)/2

)×(ϕ−(p+1+q)/(2p+q)∆ϕ

)≤ ε

∫ϕuq−2|∇u|4 + ε

∫ϕu2p+q + C(ε),

C

∫up+q|∇u · ∇ϕ| ≤

∫ (ϕ1/4u(q−2)/4|∇u|

)(ϕ(4p+3q+2)/4(2p+q) u(4p+3q+2)/4

)×(ϕ−(3p+1+2q)/2(2p+q)|∇ϕ|

)≤ ε

∫ϕuq−2|∇u|4 + ε

∫ϕu2p+q + C(ε),

and

C

∫uq−1|∇u|2|∇u · ∇ϕ| ≤

∫ (ϕ3/4u3(q−2)/4|∇u|3

)(ϕ(q+2)/4(2p+q) u(q+2)/4

)×(ϕ−(3p+1+2q)/2(2p+q)|∇ϕ|

)≤ ε

∫ϕuq−2|∇u|4 + ε

∫ϕuq+2p + C(ε).

Combining this with (8.22), we obtain

αI + δK ≤ C(n, p, q, k)ε(I + K) + C(ε).


Since α, δ > 0, by choosing ε sufficiently small, we conclude that I, K ≤ C, hence(8.8) with r = 2p + q > max(n(p− 1)/2, p).

Proof of Theorem 8.2. Let u be a positive solution of (8.2).

Assume n ≥ 2 and denote x′ = (x1, . . . , xn−1). Choose x, x ∈ Rn+ with xn = xn.

We will show u(x) = u(x) so that u depends only on xn.

Choose the origin to be the point((

x+x2

)′, 0). Given x ∈ Rn

+, set

z =x + en

|x + en|2, v(z) = |x + en|n−2u(x) =

u(x)|z|n−2

.

The function v is the Kelvin transform of u. It solves the problem

∆v + |z|γvp = 0 in D,

v = 0 on ∂D \ 0,

(8.26)

where D := B1/2(en/2) and γ := (n − 2)p − (n + 2) ≤ 0. We want to show thatv is axisymmetric about the zn axis, i.e. v = v(|z′|, zn). Choose any direction eperpendicular to the zn-axis. Without loss of generality we may assume e = e1.

en

2

λ z1

zn

D

Σ(λ)

Figure 4: Moving planes.

We shall apply the moving planes method to problem (8.26). Given λ ∈[0, 1/2), set Σ(λ) := z ∈ D : z1 > λ, zλ := (2λ − z1, z2, . . . , zn). The point zλ isthe reflection of z with respect to the hyperplane z1 = λ and Σ(λ) is called acap. We next define

w(z; λ) := v(zλ)− v(z) for z ∈ Σ(λ)

(the parameter λ will be omitted in w when no risk of confusion arises). Then

∆w = ∆v(zλ)−∆v(z) = −|zλ|γvp(zλ) + |z|γvp(z)

=(|z|γ − |zλ|γ

)vp(zλ)− |z|γ

(vp(zλ)− vp(z)

).

Since vp(zλ)− vp(z) = pξp−1w(z; λ) for some ξ = ξ(z, λ) lying between v(zλ) andv(z), we obtain

∆w + |z|γpξp−1w =(|z|γ − |zλ|γ

)vp(zλ) ≥ 0 in Σ(λ) .

The maximum principle (see Proposition 52.1) implies v > 0 in D and ∂v/∂ν < 0on ∂D \ 0, hence w ≥ 0 on Σ(λ) for λ close to 1/2.

Setµ := infµ > 0 : w ≥ 0 in Σ(λ) for all λ ≥ µ

and assume µ > 0. Then w ≥ 0 on Σ(µ) and there exist λi ∈ (0, µ), λi →µ, such that infw(z; λi) : z ∈ Σ(λi) < 0. Since w(·; λi) ≥ 0 on ∂Σ(λi), thisinfimum is attained at some qi ∈ Σ(λi) and ∇w(qi, λi) = 0. Since w(·; λi) ≥ 0in an ε-neighborhood of ∂D ∩ Σ(λi) (with ε being independent of i), we mayassume qi → q ∈ Σ(µ) \ ∂D. Continuity arguments and w ≥ 0 on Σ(µ) guaranteew(q; µ) = 0 and ∇w(q; µ) = 0, hence w(·; µ) ≡ 0 by the maximum principle.This contradicts w > 0 on z ∈ ∂Σ(µ) : z1 > µ. Consequently, µ = 0 andw(·; 0) ≥ 0 on Σ(0). A symmetric argument shows w(·; 0) ≤ 0 on Σ(0), hence vis symmetric with respect to the hyperplane e1 = 0. Since this holds for anyhyperplane containing the zn-axis, v is axially symmetric. Hence, u = u(|x′|, xn)and, consequently, u(x) = u(x).

Thus we have reduced the problem to the case n = 1. Assume that u is a positivesolution of

u′′(t) + up(t) = 0, t > 0,

u(0) = 0.

Since u is concave and positive for t > 0, it must fulfill u′ ≥ 0. Fix t1 > 0 andconsider t > t1. Then

u(t) = u(t1) + (t− t1)u′(t1) +∫ t

t1

(t− s)u′′(s) ds.

Since u′′(s) = −up(s) ≤ −up(t1), we obtain

0 < u(t) ≤ u(t1) + (t− t1)u′(t1)−12(t− t1)2up(t1),

hence

up(t1) <2u(t1)

(t− t1)2+

2u′(t1)t− t1

→ 0 as t→ +∞,

a contradiction.

We now turn to the proof of Theorem 8.1 based on moving planes.

Proof of Theorem 8.1. Due to Theorem 8.4, we may assume n ≥ 3. Let p ≤ pS

and let u be a positive classical solution of (8.1). Set

v(z) :=1

|z|n−2u( z

|z|2), z ∈ R

n \ 0

(v is the Kelvin transform of u). We have v ∈ C(Rn \ 0), v > 0,

v(z) ≤ C|z|2−n as |z| → ∞, (8.27)

and v solves the equation

∆v + |z|γvp = 0 in Rn \ 0, (8.28)

where γ := (n − 2)p − (n + 2) ≤ 0. Due to (8.28) and n ≥ 3, we infer fromLemma 4.4 that ∆v ≤ 0 in D′(Rn). It follows from the maximum principle inProposition 52.3(ii) that, for each R > 0,

v ≥ η(R) := min|x|=R

v > 0 in BR(0) \ 0. (8.29)

Given λ ≤ 0, set zλ := (2λ − z1, z2, . . . , zn), Σ(λ) := z ∈ Rn : z1 < λ,

Σ′(λ) := Σ(λ) \ 0λ and

w(z; λ) := v(zλ)− v(z), z ∈ Σ(λ) \ 0λ

(the parameter λ will be omitted in w when no risk of confusion arises). As in thepreceding theorem we obtain

∆w + |z|γpξp−1w ≤ 0 in Σ′(λ), (8.30)

where ξ = ξ(z, λ) lies between v(zλ) and v(z). Set α := (n − 2)/2 and w(z; λ) =|z|αw(z; λ). Then

∆w − n− 2|z|2 z · ∇w + c(z, λ)w ≤ 0 in Σ′(λ), (8.31)

where

c(z, λ) := − (n− 2)2

4|z|2 + |z|γpξp−1(z, λ).

Let us first show that

w ≥ 0 in Σ′(λ), for λ −1. (8.32)

We shall argue by contradiction. Assume that λi → −∞ and infΣ′(λi) w(·; λi) < 0.By (8.27) and (8.29) with R = 1, we have w(z; λi) ≥ 0 if |z − 0λi | < 1 and i islarge enough. Since also, for each i,

w(z; λi) = 0 on ∂Σ(λi) and w(z; λi)→ 0, |z| → ∞, (8.33)

we see that the infimum of w(·; λi) over Σ′(λi) is attained at some qi ∈ Σ′(λi)and |qi − 0λi | ≥ 1. We have |qi| → ∞, thus v(qi) → 0. If the sequence qλi

i were bounded, then (8.29) would imply v(qλi

i ) ≥ c1 > 0, hence w(qi) > 0 fori large, a contradiction. Therefore |qλi

i | → ∞. Now the definition of v impliesv(z)|z|n−2 → u(0) if |z| → ∞, so that we cannot have |qλi

i |/|qi| → 0 (otherwisew(qi) > 0 for large i). Thus both v(qi) and v(qλi

i ) can be estimated above byCq2−n

i for some fixed C > 0 and the same is true for ξ(qi, λi). Hence,

c(qi, λi) ≤ −(n− 2)2

4q2i

+Cp

q4i

< 0 for i large enough. (8.34)

Since w = w(·; λi) attains an interior minimum at qi, we have ∆w(qi) ≥ 0,∇w(qi) = 0 and w(qi) < 0 so that (8.31) and (8.34) yield a contradiction. Thisproves (8.32).

Now denote

µ := supµ ≤ 0 : w(·; λ) ≥ 0 in Σ′(λ) for all λ ≤ µ

and assume µ < 0. Then w(·, µ) ≥ 0 in Σ′(µ) by continuity, and there existλi > µ, λi → µ, such that infw(z; λi) : z ∈ Σ′(λi) < 0. Assume that w(·, µ)is not identically zero. Since ∆w(·, µ) ≤ 0 in Σ′(µ), the maximum principle (seeProposition 52.1) implies w(·, µ) > 0 in Σ′(µ). Arguing similarly as for (8.29), wededuce that w(·, µ) ≥ c2 > 0 in U := Bµ/2(0µ) \ 0µ. Due to the continuity of vin U and

w(z; λi) = w(z − 2(λi − µ)e1; µ

)+v(z − 2(λi − µ)e1

)−v(z), e1 := (1, 0, . . . , 0),

we obtain w(·; λi) ≥ 0 (hence w(·; λi) ≥ 0) in Bµ/4(0λi) \ 0λi for i large. Conse-quently, in view of (8.33), the infimum of w(·; λi) over Σ′(λi) has to be attained atsome qi ∈ Σ′(λi), with |qi−0λi | ≥ µ/4. Assume |qi| → ∞. Then |qλi

i |/|qi| → 1 andwe obtain a contradiction as above (cf. (8.34)). Therefore we may assume that qiis bounded and qi → q ∈ Σ(µ) \ 0µ. By continuity and w(·, µ) ≥ 0, we obtainw(q, µ) = 0 and ∇w(q, µ) = 0, hence w(q, µ) = 0 and ∇w(q, µ) = 0. Applying themaximum principle in Proposition 52.1(ii) and (iii) to equation (8.30), it followsthat w(·, µ) ≡ 0, hence w(·, µ) ≡ 0, a contradiction. Consequently, w(·, µ) ≡ 0,which means that v is symmetric with respect to z1 = µ. Now using (8.28) wesee that (−∆v)/vp = |z|γ has the same symmetry, which is not possible unlessp = pS .

If p < pS , then we get µ = 0, so that w(·, 0) ≥ 0 and v(z0) ≥ v(z) providedz1 ≤ 0. Considering the function v(z) := v(z0) instead of v we obtain the reversedinequality, hence v(z1, z2, . . . , zn) = v(−z1, z2, . . . , zn). Repeating this procedurewith any given direction instead of e1 we see that v, hence u, are radially symmetric(about zero). If we repeat this procedure with u(x) = u(x− x0) for a given x0 = 0instead of u, we show that u is radially symmetric about the point x0. Since this istrue for any x0, the function u has to be constant. But the only constant solutionof (8.1) is the trivial solution.

If p = pS and µ < 0, then v is symmetric with respect to z1 = µ. If µ = 0,then we can repeat the procedure with v(z) := v(z0) and in any case we obtainthe symmetry of v with respect to z1 = µ for suitable µ. Now we can repeat theabove proof with directions e2, e3, . . . , en instead of e1 and we obtain the existenceof z ∈ R

n such that v is symmetric with respect to zk = zk for k = 1, 2, . . . , n,hence v(z + z) = v(z − z) for all z. Rotating the coordinate system and repeatingthe procedure we find z ∈ R

n such that v(z+z) = v(z−z) for all z. Assume z = z.Without loss of generality, we may assume z = 0. The symmetry relations for vimply

v(z) = v(2z − z) = v(3z − 2z) = v(4z − 3z) = · · · → 0,

hence v(z) = 0, a contradiction. Consequently, z = z and we obtain the rotationalsymmetry of v (hence of u) about z.

Proof of Theorem 8.3. Assume that (8.2) admits a positive, bounded classicalsolution u. As a special case of Theorem 21.10 below (which we shall prove byusing moving planes arguments), it follows that u is nondecreasing in xn:

∂xnu(x) ≥ 0, x ∈ Rn+.

Therefore, for each x′ ∈ Rn−1,

U(x′) := limxn→∞u(x′, xn)

is well defined and is a bounded positive function. Take now ϕ ∈ D(Rn−1) andψ ∈ D(R), with supp ψ ⊂ (0, 1) and

∫ 1

0ψ = 1. Let k > 0. Testing the equation

with ϕ(x′)ψ(xn − k), we have

−∫

Rn−1

∫R

up ϕ(x′)ψ(xn − k) dxn dx′ =∫

Rn−1

∫R

ϕ(x′)ψ(xn − k)∆u dxn dx′

=∫

Rn−1

∫R

u ∆(ϕ(x′)ψ(xn − k)

)dxn dx′,

hence

−∫

Rn−1

∫R

up(x′, s + k)ϕ(x′)ψ(s) ds dx′

=∫

Rn−1

∫R

u(x′, s + k)∆(ϕ(x′)ψ(s)

)ds dx′.

By dominated convergence, letting k →∞, it follows that

−∫

Rn−1Up(x′)ϕ(x′) dx′ = −

∫Rn−1

Up(x′)ϕ(x′)∫ 1

0

ψ(s) ds dx′

=∫

Rn−1

∫R

U(x′)∆(ϕ(x′)ψ(s)

)ds dx′.

But the RHS is equal to∫Rn−1

U(x′)∆x′ϕ(x′) dx′∫ 1

0

ψ(s) ds +∫

Rn−1U(x′)ϕ(x′) dx′

∫ 1

0

ψ′′(s) ds

=∫

Rn−1U(x′)∆x′ϕ(x′) dx′.

It follows that U solves (8.1) in Rn−1 in the distribution sense, hence in the classical

sense (this is a consequence of the boundedness of U and of Remark 47.4). Theresult is then a consequence of Theorem 8.1(i).

9. Positive radial solutions of ∆u + up = 0 in Rn

In this section we study positive radial classical solutions of the equation

−∆u = up, x ∈ Rn. (9.1)

Since this problem does not possess positive classical solutions if 1 < p < pS dueto Theorem 8.1, we restrict ourselves to the case p ≥ pS . Consequently, n ≥ 3.

Positive radial classical solutions of (9.1) can be written in the form u(x) = U(r),where r = |x| and U ∈ C2([0,∞)) is a positive classical solution of

U ′′ +n− 1

rU ′ + Up = 0, r ∈ (0,∞), U ′(0) = 0. (9.2)

It is easily seen that prescribing initial values U(0) = α > 0, U ′(0) = 0, theequation in (9.2) has a unique solution for r small enough. In fact, this equationcan be written in the form (rn−1U ′)′ = −rn−1Up and, by integration we obtainthe equivalent integral equation

U(r) = α−∫ r

0

∫ s

0

( t

s

)n−1

Up(t) dt ds,

which can be solved by the Banach fixed point theorem.Let U∗(r) = cpr

−2/(p−1) be the singular solution defined in (3.9) and set

pJL :=

+∞ if n ≤ 10,

1 + 4n−4+2√

n−1(n−2)(n−10) if n > 10.

(9.3)

The main result of this section is the following theorem.

9. Positive radial solutions of ∆u+up=0 in Rn 51

Theorem 9.1. Let p ≥ pS. Given α > 0, problem (9.2) possesses a unique positivesolution Uα ∈ C2([0,∞)) satisfying Uα(0) = α. This solution is decreasing and wehave

Uα(r) = αU1(α(p−1)/2r). (9.4)

If p > pS, then r2/(p−1)Uα(r) → cp as r →∞. If p = pS, then

U1(r) =( n(n− 2)

n(n− 2) + r2

)(n−2)/2

. (9.5)

Let α1 > α2 > 0. If p ≥ pJL, then U∗(r) > Uα1(r) > Uα2(r) for all r > 0.If pS < p < pJL, then Uα1 and Uα2 intersect infinitely many times and Uα1 , U∗intersect infinitely many times as well. If p = pS, then Uα1 , Uα2 intersect once andUα1 , U∗ intersect twice.

Proof. Using the transformation

w(s) = r2/(p−1)U(r), s = log r, (9.6)

problem (9.2) becomes

w′′ + βw′ + wp − γw = 0, s ∈ R, (9.7)

where

β :=1

p− 1((n− 2)p− (n + 2)

)≥ 0, γ := cp−1

p =2

(p− 1)2((n− 2)p− n

)> 0,

and we are looking for solutions w satisfying w(s), w′(s) → 0 as s → −∞. Set

E(w) = E(w, w′) :=12|w′|2 − γ

2w2 +

1p + 1

wp+1.

Then E is a Lyapunov functional for (9.7); more precisely,

d

dsE(w(s)

)= −β

(w′(s)

)2≤ 0. (9.8)

Denoting x := w and y := w′, problem (9.7) can be written in the form(x

y

)′=( y

−βy − xp + γx

)=: F (x, y) (9.9)

where x > 0 and (x, y)→ (0, 0) as s → −∞. Problem (9.9) possesses two equilibria,(0, 0) and (cp, 0) lying in the half-space (x, y) : x ≥ 0. Denote

A1 := ∇F (0, 0) =( 0 1

γ −β

), A2 := ∇F (cp, 0) =

( 0 1−γ(p− 1) −β

).

0

y

x

y = ν2x

y′ = 0

T

cp

Figure 5: The flow generated by (9.9) for pS pS . Then β > 0 and the matrix A1 has two realeigenvalues ν1,2 := − 1

2

(β ±

√β2 + 4γ

)with ν1 < 0 < ν2 = 2/(p − 1). The

corresponding eigenvectors (xi, yi) satisfy yi = νixi, i = 1, 2. The eigenvaluesν1,2 := − 1

2

(β ±

√β2 − 4γ(p− 1)

)of A2 are real iff β2 ≥ 4γ(p − 1), that is iff

p ≥ pJL.

Assume pS < p < pJL. In this case, the eigenvalues ν1, ν2 are complex and theirreal parts are negative so that the critical point (cp, 0) is a stable spiral. The flowfor the planar system (9.9) is illustrated in Figure 5.

We are interested in the trajectory T emanating from the origin to the righthalf-space, since it represents the graph of any positive solution of (9.7) in the w-w′

plane. This trajectory cannot hit the axis x = 0 again since the energy functionalE is nonnegative on this axis, E(0, 0) = 0, β > 0 and (9.8) is true. Moreover,the corresponding solutions w exists for all s ∈ R and w, w′ remain boundedfor all s ∈ R due to (9.8). Consequently, T has to converge to the critical point(cp, 0) which corresponds to the singular solution w∗(s) = r2/(p−1)U∗(r) ≡ cp.Thus, if Uα is the unique local solution of (9.2) such that Uα(0) = α > 0, thenits transform wα(s) = r2/(p−1)Uα(r) exists globally and satisfies wα(s) → cp ass → ∞. Consequently, Uα exists globally and r2/(p−1)Uα(r) → cp as r → ∞. Itis easily verified that the function Uα(r) := αU1(α(p−1)/2r) is a solution of (9.2)satisfying Uα(0) = α, hence Uα = Uα by uniqueness. The graphs of wα and w1

in the w-w′ plane are identical, so that there exists sα ∈ R such that Uα(es) =wα(s) = w1(s − sα) for all s ∈ R. Hence, given α1 > α2 > 0, Uα1(r) = Uα2(r)for some r > 0 iff w1(s − sα1) = w1(s − sα2) for some s ∈ R. This happens forinfinitely many s since T spirals around the point (cp, 0). Similarly, wα1(s) = cp

9. Positive radial solutions of ∆u+up=0 in Rn 53

for infinitely many s, hence Uα1 and U∗ intersect infinitely many times.

Next consider the case p ≥ pJL. On the halfline y = −β2 (x − cp), x < cp, we

have for suitable xθ ∈ (x, cp):

y′

x′ = −β − x

y(xp−1 − γ) = −β +

2x(xp−1 − cp−1p )

β(x − cp)

= −β +2β

x(p− 1)xp−2θ < −β +

2β

(p− 1)γ ≤ −β

2.

0

y

x

y = ν2x

y = −β2 (x− cp)

y′ = 0

T

cp

Figure 6: The flow generated by (9.9) for p ≥ pJL.

Consequently, the trajectory T ends up at (cp, 0) again but the x-coordinateis increasing along T (see Figure 6). Hence, the solutions U of (9.2) are orderedaccording to their values at r = 0, U∗ > Uα1 > Uα2 if α1 > α2.

Finally consider the case p = pS . Then β = 0 and the energy functional E isconstant along any solution. Since E(cp, 0) < 0 and E(0, y) > 0 for y = 0, thetrajectory T is a homoclinic orbit (see Figure 7).

Let wα, sα have the same meaning as above. Given α1 = α2, there exists aunique s ∈ R such that w1(s − sα1) = w1(s − sα2). Hence, the correspondingsolutions Uα1 , Uα2 of (9.2) intersect exactly once. Similarly, given α > 0, we havewα(s) = cp for two values of s, so that Uα and U∗ intersect twice. One can easilycheck that the function U1 defined by (9.5) is a solution of (9.2) satisfying theinitial condition U1(0) = 1.

Remarks 9.2. (i) The exponent pJL appeared for the first time in [293] wherethe authors studied mainly problems with the nonlinearities f(u) = λ(1 + au)p

and f(u) = λeu, λ, a > 0.

y

x

T

cp

Figure 7: The flow generated by (9.9) for p = pS .

(ii) The intersection properties of the solutions U in Theorem 9.1 play an im-portant role in the study of stability and asymptotic behavior of solutions of thecorresponding parabolic problem, see Sections 22, 23.

Remark 9.3. Let p = pS and a > 0. For all α ≥ M0(a) with M0(a) > 0 largeenough, if V is a positive classical solution of

V ′′ +n− 1

rV ′ + V p = 0, 0 < r < a,

such that V (a) = Uα(a) and limr→0 V (r) = ∞, then V has to intersect Uα in(0, a).

In fact, denoting wα(s) := r2/(p−1)Uα(r), s = log r, the rescaled function fromthe last proof, it suffices to chose M0(a) such that

w′M0(a)(log a) < 0 (9.10)

(hence w′α(log a) < 0 for all α ≥ M0(a)). Indeed the trajectory of W (s) :=

r2/(p−1)V (r), s ∈ (−∞, log a), has to be a subset of a periodic orbit lying in-side the trajectory T (see Figure 7). Due to (9.10) there exists s0 ∈ (−∞, log a)such that wα(s0) = W (s0), hence Uα(es0) = V (es0).

Note also that there exist infinitely many periodic orbits of (9.7) for p = pS ,corresponding to positive singular solutions of u′′ + n−1

r u′ + up = 0 for r > 0.

Remark 9.4. Let p > pJL. Since the trajectory T approaches the limit point(cp, 0) below the dotted line with slope −β/2 and ν2 < −β/2 < ν1 < 0, it has toconverge along the eigenvector (1, ν1) corresponding to the eigenvalue ν1, hence

y(s)x(s) − cp

→ ν1 as s →∞.

10. A priori bounds via the method of Hardy-Sobolev inequalities 55

Returning to the original variables and denoting V (r) := U(r) − U∗(r) we obtain

limr→∞

rV ′(r)V (r)

= ν1 −m, (9.11)

where m := 2/(p − 1). Assuming that V (r) = cr−α + h.o.t. for some c = 0 andα > m, (9.11) guarantees c < 0 and α = m + λ−, where

λ− := −ν1 =12(β −

√β2 − 4γ(p− 1)

)=

12(n− 2− 2m−

√(n− 2− 2m)2 − 8(n− 2−m)

).

This expansion is indeed true: In fact, a more precise asymptotic expansion of Vwas established in [260] and [334].

10. A priori bounds via the method ofHardy-Sobolev inequalities

A priori estimates of solutions can be used for the proof of existence and multi-plicity results. Unlike the variational methods in sections 6 and 7, this approachdoes not require any variational structure of the problem and enables one to provethe existence of continuous branches of solutions.

Due to Theorem 7.8(ii) one cannot hope for a priori estimates of all solutions.The bifurcation diagrams in Figure 2 suggest that there is some hope for suchestimates if we restrict ourselves to positive solutions and to the subcritical case.3

In the present and the following three sections we introduce four different meth-ods which are often used in the proofs of a priori bounds for positive solutions ofsuperlinear elliptic problems. We will study mainly the scalar problem

−∆u = f(x, u,∇u), x ∈ Ω,

u = 0, x ∈ ∂Ω,

(10.1)

where Ω is bounded and f is a sufficiently smooth function with superlinear growthin the u-variable. Some of the possible generalizations and modifications will bementioned as remarks, others can be found in the subsequent chapters.

This section is devoted to the method of [99], which is based on a Hardy-type inequality and enables one to treat rather general nonlinearities f . On the

3In fact, in the subcritical case one can get a priori estimates of all solutions with boundedMorse indices (without the positivity assumption), see [49], [539], [32].

other hand, it requires an upper growth restriction corresponding to the limitingexponent

pBT := ∞ if n = 1,

(n + 1)/(n− 1) if n > 1,

which is stronger than what is imposed by the methods in Sections 12 and 13 (forinstance, in the particular case f(x, u,∇u) = up, we have to assume p < pBT ).However, the exponent pBT is not technical and its role will be clarified in thenext section.

Theorem 10.1. Let Ω ⊂ Rn be bounded, n ≥ 3, β := pBT . Let f : Ω×R+×R

n →R+ be continuous and bounded on Ω×M × R

n for M ⊂ R+ bounded. Let

lim infu→∞

f(x, u, s)u

> λ1, limu→∞

f(x, u, s)uβ

= 0, uniformly for (x, s) ∈ Ω× Rn.

(10.2)Then there exists C > 0 with the following property: If t ≥ 0 and u ∈ H1

0 ∩L∞(Ω)is a positive variational solution of

−∆u = f(x, u,∇u) + tϕ1, x ∈ Ω,

u = 0, x ∈ ∂Ω,

(10.3)

then‖u‖∞ + t ≤ C. (10.4)

Proof. We shall denote by C various positive constants which may vary fromstep to step but which are independent of u and t. Let t ≥ 0 and u be a positivesolution of (10.3). The proof of (10.4) will consist of the following three steps:

1.∫Ω uδ dx ≤ C, t ≤ C and

∫Ω f(x, u,∇u)δ dx ≤ C,

2. ‖∇u‖2 ≤ C,3. ‖u‖∞ ≤ C.

Step 1. Due to (10.2) there exist C1 > λ1 and C2 > 0 such that f(x, u, s) ≥C1u− C2 for all (x, u, s). Multiplying the equation in (10.3) by ϕ1 yields

λ1

∫Ω

uϕ1 dx =∫

Ω

u(−∆ϕ1) dx =∫

Ω

(−∆u)ϕ1 dx =∫

Ω

(fϕ1 + tϕ21) dx

≥ C1

∫Ω

uϕ1 dx− C2

∫Ω

ϕ1 dx + t

∫Ω

ϕ21 dx,

(10.5)

where f = f(x, u(x),∇u(x)). This estimate can be written in the form

(C1 − λ1)∫

Ω

uϕ1 dx + t

∫Ω

ϕ21 dx ≤ C,

hence ∫Ω

uϕ1 dx ≤ C and t ≤ C. (10.6)

Now (10.5) and δ ≤ Cϕ1 guarantee∫Ω

fδ dx ≤ C

∫Ω

fϕ1 dx = Cλ1

∫Ω

uϕ1 dx− Ct

∫Ω

ϕ21 dx ≤ C. (10.7)

Step 2. Multiplying the equation in (10.3) by u yields

‖∇u‖22 =∫

Ω

|∇u|2 dx =∫

Ω

fu dx + t

∫Ω

ϕ1udx ≤∫

Ω

fu dx + C. (10.8)

Denoting α := 2/(n + 1) ∈ (0, 1) we have β + 1/(1− α) = 2/(1− α). Given ε > 0there exists Cε > 1 such that

f(x, u, s) ≤ εuβ + Cε. (10.9)

Using Holder’s inequality, Step 1, (10.9) and Lemma 50.4 we obtain∫Ω

fu dx =∫

Ω

(fαδα)(f1−α u

δα

)dx ≤

(∫Ω

fδ dx)α(∫

Ω

fu1/(1−α)

δα/(1−α)dx)1−α

≤ ε1−α(∫

Ω

uβ+1/(1−α)


+ Cε

(∫Ω

u1/(1−α)


= ε1−α∥∥∥ u

δα/2

∥∥∥2

2/(1−α)+ Cε

∥∥∥ u

δα

∥∥∥1/(1−α)

≤ ε1−αC‖∇u‖22 + CCε‖∇u‖2.

This estimate and (10.8) guarantee

‖∇u‖2 ≤ C. (10.10)

Step 3. Choose p ∈ (n/2, n). Then

W 2,p(Ω) → L∞(Ω) and W 1,2(Ω) → Lp(β−1)(Ω)

due to n(β − 1) < 2∗. These embeddings, Lp-estimates (see Appendix A), (10.9),Step 1 and (10.10) imply

‖u‖∞ ≤ C‖u‖2,p ≤ C‖f + tϕ1‖p ≤ ε‖uβ‖p + C(Cε + 1)

≤ ε‖u‖β−1p(β−1)‖u‖∞ + Cε ≤ ε‖∇u‖β−1

2 ‖u‖∞ + Cε ≤ εC‖u‖∞ + Cε.

Now choosing ε > 0 small enough yields ‖u‖∞ < C.

Remarks 10.2. (i) The proof of Theorem 10.1 can be easily modified for moregeneral second-order elliptic differential operators. In the case of a nonsymmetricoperator one has to work with the first eigenfunction of the adjoint operator, ofcourse. One could also allow more general nonlinearities (nonlocal, for example).The boundedness assumption on f could be relaxed as well.

(ii) The term tϕ1 in (10.3) is needed for the proof of existence of a positivesolution of (10.3) with t = 0 (see Corollary 10.3 below). This lower order term doesnot play any significant role in a priori estimates in the following sections providedt ≤ C. Since this bound for t was proved in Step 1 of the proof of Theorem 10.1by using only the lower bound for f in (10.2), in the following sections we shallrestrict ourselves to the case t = 0 only.

(iii) A priori estimates of solutions of problems like (10.3) appeared first in [400]and [517]. The assumptions on the growth of f or the dimension n in these articlesare more restrictive than those in Theorem 10.1 which is due to [99].

Corollary 10.3. Let Ω and f be as in Theorem 10.1 and let

lim supu→0+

f(x, u, s)u

< λ1 uniformly for (x, s) ∈ Ω× Rn. (10.11)

Then problem (10.3) with t = 0 possesses at least one positive solution u, withu ∈W 2,q ∩ C0(Ω) for all finite q.

Proof. Set X := C1(Ω). Given u ∈ X and t ≥ 0, let Ft(u) = w be the uniquesolution of the linear problem

−∆w = f(x, u,∇u) + tϕ1, x ∈ Ω,

w = 0, x ∈ ∂Ω

(10.12)

(cf. Theorem 47.3(i)). Note that, since f(·, u,∇u) ∈ L∞(Ω), we have u ∈ W 2,q ∩C0(Ω) for all finite q. Then Ft : X → X is compact and we are looking for apositive fixed point of F0.

Let ‖u‖X = r 1, τ ∈ [0, 1] and assume τF0(u) = u. Multiplying the equationin (10.12) by u and applying (10.11) yield∫

Ω

|∇u|2 dx = τ

∫Ω

fu dx ≤ (λ1 − ε)∫

Ω

u2 dx,

which contradicts (1.3). Hence τF0(u) = u and the homotopy invariance of thetopological degree implies

deg(I − F0, 0, Br

)= deg

(I, 0, Br

)= 1, (10.13)

where I denotes the identity and Br := u ∈ X : ‖u‖X < r.

Let ‖u‖X = R. If R is large enough, then Theorem 10.1 and Lp-estimates (seeAppendix A) imply Ft(u) = u for any t ≥ 0. The same theorem implies alsoFT (u) = u provided T is large enough. Consequently,

deg(I − F0, 0, BR

)= deg

(I − FT , 0, BR

)= 0. (10.14)

Now (10.13) and (10.14) guarantee deg(I − F0, 0, BR \ Br

)= −1, hence there

exists u ∈ BR \ Br such that F0(u) = u. The positivity of u is a consequence ofthe maximum principle.

In what follows we present an alternative proof of Theorem 10.1 in the specialcase f(x, u, s) = |u|p−1u, 1 < p < pBT , n ≥ 1. Instead of Hardy’s inequality weshall use the following lemma (see [89], [450], and cf. also [143] and the referencesin [450, Remark 4.1]). It provides a useful singular test-function and will also beused later in Section 26.

Lemma 10.4. Assume Ω bounded and 0 < α < 1. Then the problem

−∆ξ = ϕ−α1 , x ∈ Ω,

ξ = 0, x ∈ ∂Ω

(10.15)

admits a unique classical solution ξ ∈ C(Ω) ∩ C2(Ω). Moreover, we have ϕ−α1 ∈

L1(Ω), ξ ∈ H10 (Ω), and

ξ(x) ≤ C(Ω, α)δ(x), x ∈ Ω. (10.16)

Proof. Define h(s) = 3s− s2−α, s ≥ 0. The function h ∈ C1([0,∞))∩C2((0,∞))satisfies

h′ = 3− (2− α)s1−α, −h′′ = (2 − α)(1− α)s−α, s > 0

andh(s) ≤ 3s, h′(s) ≥ 1, for all s ∈ [0, 1].

Let ϕ = ‖ϕ1‖−1∞ ϕ1, and set v(x) = h(ϕ(x)). Simple computation yields

−∆v = −h′′(ϕ)|∇ϕ|2 − h′(ϕ)∆ϕ

= C1ϕ−α|∇ϕ|2 + λ1h

′(ϕ)ϕ

≥ C1ϕ−α|∇ϕ|2 + λ1ϕ.

Now, for δ(x) ≤ ε small enough, we have |∇ϕ|2 ≥ η > 0, hence −∆v ≥ C1ηϕ−α.On the other hand, for δ(x) ≥ ε, we have ϕ ≥ c > 0, hence −∆v ≥ λ1c ≥ C2ϕ

−α.We conclude that for some c > 0, w := cv satisfies

−∆w ≥ ϕ−α1 and w(x) ≤ C3δ(x), for all x ∈ Ω. (10.17)

Next, for all ε > 0, let ξε be the (classical) solution of −∆ξε = (ϕ1 + ε)−α in Ω,with ξε = 0 on ∂Ω. By (10.17) and the maximum principle, we have

ξε(x) ≤ w(x) ≤ C3δ(x) ≤ C4, x ∈ Ω (10.18)

and ξε is increasing as ε decreases to 0. Denote by ξ the (pointwise) limit of ξε.Elliptic estimates along with (10.18) imply that ξ ∈ C(Ω)∩C2(Ω), that ξ satisfies(10.16) and is a classical solution of (10.15). The uniqueness follows immediatelyfrom the maximum principle.

The fact that ϕ−α1 ∈ L1(Ω) can be easily deduced from the inequality ϕ1 ≥ cδ,

by flattening the boundary and using a partition of unity (see e.g. [485] for details).Finally, to show that ξ ∈ H1

0 (Ω), it suffices to note that, since α < 1,∫Ω

|∇ξε|2 = −∫

Ω

ξε∆ξε =∫

Ω

ξε(ϕ1 + ε)−α ≤ C4

∫Ω

ϕ−α1 < ∞.

Alternative proof of Theorem 10.1 for f = up, t = 0. Let ε > 0 be smalland α := r′/r, where r is defined by 1/r = 1/2− ε/(p− 1). Let ξ be the solutionof (10.15). As in Step 1 of the proof of Theorem 10.1 we obtain

∫Ω upδ dx ≤ C.

Testing the equation with ξ, we obtain∫Ω

uϕ−α1 dx =

∫Ω

∇u · ∇ξ dx =∫

Ω

(−∆u)ξ dx =∫

Ω

upξ dx ≤ C

(where we used ϕ−α1 ∈ L1(Ω) and ξ ∈ H1

0 (Ω)). Denoting pε := (p + 1)/2 − ε, weget ∫

Ω

upε dx =∫

Ω

(up/rϕ

1/r1

)(u1/r′

ϕ−1/r1

)dx

≤(∫

Ω

upϕ1 dx)1/r(∫

Ω

uϕ−α1 dx

)1/r′

≤ C.

Define θ ∈ (0, p + 1) by θ/pε + (p + 1− θ)/2∗ = 1. Then p + 1− θ < 2 provided εis small enough and the interpolation inequality yields∫

Ω

|∇u|2 dx =∫

Ω

up+1 dx = ‖u‖p+1p+1 ≤ ‖u‖θ

pε‖u‖p+1−θ

2∗ ≤ C‖∇u‖p+1−θ2 ,

which guarantees a bound for u in W 1,2(Ω). The rest of the proof is the same asin the proof of Theorem 10.1 (Step 3).

11. A priori bounds via bootstrap in Lpδ-spaces 61

11. A priori bounds via bootstrap in Lpδ-spaces

This section is devoted to the Lpδ bootstrap method, which, in the scalar case,

was developed independently in [85], [449]. It applies to problem (10.1) underessentially the same assumptions on the nonlinearities f as in the method of theprevious section, with a growth restriction still given by the exponent pBT ofSection 10. However, unlike that method (and those in the next two sections), itapplies to very weak solutions. The optimality of the Lp

δ bootstrap method wasstudied in [489] and it turns out that the exponent pBT is optimal for the regularityof very weak solutions, thus showing the critical role played by this exponent forproblems of the form (10.1).

Let us point out that in the case of systems, studied in [449], the growth restric-tions of the Lp

δ bootstrap method become much weaker than those imposed by the(generalization of the) method of Hardy-Sobolev inequalities (see Section 31).

In this section, by a solution u of (10.1), we understand a very weak (or L1δ-)

solution, cf. Definition 3.1. Namely, if f does not depend on ∇u, this means that

u ∈ L1(Ω), f(·, u) ∈ L1δ(Ω), (11.1)

and−∫

Ω

u∆ϕ =∫

Ω

f(·, u)ϕ, for all ϕ ∈ C2(Ω), ϕ|∂Ω = 0. (11.2)

If f depends on ∇u, we assume in addition that∇u is a function, i.e.∇u ∈ L1loc(Ω)

and we replace f(·, u) by f(·, u,∇u) in (11.1)–(11.2).

Remark 11.1. If u ∈ L1(Ω) and ∆u ∈ L1δ(Ω) (where ∆u is understood in the

distribution sense), we say that u = 0 on ∂Ω in the weak sense if∫Ω

u ∆ϕ =∫

Ω

ϕ∆u for all ϕ ∈ C2(Ω), ϕ|∂Ω = 0.

If (11.1) is satisfied (and ∇u ∈ L1loc(Ω) in case f depends on ∇u), then u is a very

weak solution of (10.1) if and only if it solves the differential equations in (10.1)in the distribution sense and the boundary conditions in the weak sense.

Theorem 11.2. Assume Ω bounded and 1 λ1. (11.4)

There exists C > 0 such that if u is a nonnegative very weak solution of (10.1),then u ∈ L∞(Ω) and

‖u‖∞ ≤ C.

Condition (11.4) can be weakened or replaced by other conditions of differentform. For instance, by applying the same method, we obtain regularity and a prioriestimates for the following simple equation:

−∆u = a(x)up, x ∈ Ω,

u = 0, x ∈ ∂Ω.

(11.5)

Theorem 11.3. Assume Ω bounded and a ∈ L∞(Ω), a ≥ 0, a ≡ 0 and 1 < p <pBT . Then the conclusions of Theorem 11.2 remain valid for problem (11.5).

Remarks 11.4. (i) The growth condition (11.3) in Theorem 11.2 is slightlystronger than that in Theorem 10.1 (where (10.2) allows some “almost critical”f ’s).

(ii) Under the assumptions of Theorems 11.2 and 11.3, as a consequence ofstandard regularity results for linear elliptic equations, we moreover obtain u ∈C0 ∩W 2,q(Ω) for all finite q (argue similarly as in the proof of Corollary 3.4, usingthe uniqueness part of Theorem 49.1 instead of Proposition 52.3).

The optimality of the exponent pBT in Theorems 11.2 and 11.3 is shown by thefollowing result from [489].

Theorem 11.5. Assume Ω bounded and p > pBT . Then there exists a functiona ∈ L∞(Ω), a ≥ 0, a ≡ 0, such that problem (11.5) admits a positive very weaksolution u such that

u ∈ L∞(Ω).

The method of proof of Theorems 11.2–11.3 is based on bootstrap and uses theLp

δ regularity theory of the Laplacian (cf. Theorem 49.2 and Proposition 49.5 inAppendix C).

Proof of Theorem 11.2. Step 1. Initialization. By (10.6), (10.7) in the proof ofTheorem 10.1, we know that

‖u‖1,δ ≤ C, ‖f(·, u,∇u)‖1,δ ≤ C. (11.6)

Since p < pBT , we may fix ρ > 1 and k0 such that

max(p,

n + 12

(p− 1

ρ

))< k0 <

n + 1n− 1

.

By (11.6) and Proposition 49.5, it follows that ‖u‖k0,δ ≤ C.

11. A priori bounds via bootstrap in Lpδ-spaces 63

Step 2. Bootstrap. Put ki = k0ρi, i = 1, 2, . . . . Assume that there holds

‖u‖ki,δ ≤ C(i) (11.7)

for some i ≥ 0 (this is true for i = 0 by Step 1). Since

p

ki− 1

ki+1=

1k0ρi

(p− 1

ρ

)<

2n + 1

,

by using Theorem 49.2(i) and (11.3), we obtain

‖u‖ki+1,δ ≤ C‖∆u‖ki/p,δ = C‖f‖ki/p,δ

≤ C(1 + ‖vp‖ki/p,δ) = C(1 + ‖v‖pki,δ

) ≤ C.

By induction, it follows that (11.7) is true for all integers i. Taking i large enough,we thus have (11.7) for some ki > (n+1)p/2. Applying Theorem 49.2(i) and (11.3)once more, and Remark 1.1, we obtain ‖u‖∞ ≤ C.

Proof of Theorem 11.3. We only need to modify Step 1, the bootstrap stepbeing then unchanged.

Assume that u is a nonnegative (very weak) solution of (11.5). It follows fromthe quantitative version of Hopf’s lemma (see Remark 49.12(i) in Appendix C)that

u ≥ c(∫

Ω

aupδ dy)

δ ≥ c1

(∫Ω

aupϕ1 dy)

ϕ1,

for some constant c1 > 0 depending only on Ω. We deduce that∫Ω

aupϕ1 dx ≥ cp1

(∫Ω

aupϕ1 dx)p∫

Ω

aϕp+11 dx ≥ 2

∫Ω

aupϕ1 dx− C,

henceλ1

∫Ω

uϕ1 dx =∫

Ω

aupϕ1 dx ≤ C.

We now turn to the proof of Theorem 11.5. It is based on Lemma 49.13 fromAppendix C, where a singular solution of the linear Laplace equation with anappropriate right-hand side belonging to L1

δ is constructed. The right-hand sidehas to possess suitable boundary singularities, supported in a conical subdomainof Ω. In order to re-construct a posteriori the coefficient a(x), the key point is thelower estimate (11.8) for the solution in the same cone.

Proof of Theorem 11.5. Assume that 0 ∈ ∂Ω without loss of generality. Letα = 2/(p− 1). By assumption, we have α < n − 1. By Lemma 49.13, there existR > 0 and a revolution cone Σ1 of vertex 0, with Σ := Σ1 ∩ B2R ⊂ Ω, such thatthe function

φ := |x|−(α+2)χΣ

belongs to L1δ and such that the (very weak) solution u > 0 of

−∆u = φ, x ∈ Ω,

u = 0, x ∈ ∂Ω

satisfies

u ≥ C|x|−αχΣ. (11.8)

Therefore, we have u ∈ L∞ and

up ≥ C|x|−αpχΣ = C|x|−(α+2)χΣ = Cφ.

Setting a(x) = φ/up ≥ 0, we get −∆u = φ = a(x)up and a(x) ≤ 1/C, hencea ∈ L∞. The proof is complete.

Remarks 11.6. Localization of singularities. (a) In Theorem 11.5, it is to benoted that, in spite of the imposed homogeneous Dirichlet boundary condition,the singularity of the solution occurs at a boundary point, actually a single point.The boundary conditions continue to be satisfied not only in the weak sense butalso in the sense of traces (see Remark 49.4(c) in Appendix C).

(b) If we assume that p < psg and that a given weak solution of (11.5) is boundednear the boundary, then one can use usual Lebesgue spaces instead of Lp

δ-spaces inthe proof of Theorem 11.2, to show that the solution is bounded in Ω. Therefore,the occurrence of boundary singularities is necessary if pBT psg, the situation is different and much easier, since it isthen not difficult to construct examples of similar equations with only an interiorsingularity (see Remarks 3.6).

(c) The support of a in Theorem 11.5 can be localized in an arbitrarily smallneighborhood of a boundary point. However, it is also possible to construct anexample where the function a is positive in Ω, uniformly away from ∂Ω (see [489]for details).

Remarks 11.7. (a) The cases f(u) = up and p = pBT . Similar counter-examples as in Theorem 11.5 have been constructed recently in [155] for the modelproblem (3.10) (a(x) ≡ 1) when p > pBT is close to pBT . Moreover the criticalcase p = pBT was shown to belong to the singular case. Related results have alsobeen announced in [81].

(b) Variable critical exponents in nonsmooth domains. The notion ofvery weak solution has been recently extended in [362] to the case of some non-smooth domains, namely Lipschitz domains, and generalizations of Theorems 11.2and 11.5 have been obtained. For suitable cone-shaped domains, the analogue ofthe exponent pBT was computed. Interestingly, it was found to depend on thedomain and to be smaller than (n + 1)/(n− 1).

12. A priori bounds via the rescaling method 65

12. A priori bounds via the rescaling method

In this section we present a priori estimates of solutions of (10.3) based on rescalingand Liouville-type theorems. In this context, this method was first used in [241].In comparison to the method of Section 10, it requires a rather precise asymptoticbehavior for f as u → ∞ (f has to behave like up for u large) but the growthcondition on f is optimal (p < pS). The method also works for general second-orderelliptic operators but for simplicity we restrict ourselves to the Laplace operator.As explained in Remark 10.2(ii) we consider the case t = 0 only.

Theorem 12.1. Assume Ω bounded, 1 0 forall x ∈ Ω, g ∈ C(Ω× R× R

n), and

|g(x, u, s)| ≤ C(1 + |u|q + |s|r

), where q < p, r <

2p

p + 1. (12.1)

Then there exists C > 0 such that any positive strong solution u ∈ C1(Ω) of

−∆u = a(x)up + g(x, u,∇u), x ∈ Ω,

u = 0, x ∈ ∂Ω

(12.2)

satisfies ‖u‖∞ ≤ C.

Remark 12.2. Here, u being a strong solution means that u ∈ W 2,1loc (Ω) and

u satisfies the differential equation a.e. in Ω. Since we also assume u ∈ C1(Ω),Remarks 47.4(i) and (iii), actually imply u ∈W 2,q(Ω) for all finite q.

Proof of Theorem 12.1. Assume the contrary. Then there exist positive solu-tions uj of (12.2) such that ‖uj‖∞ →∞ as j →∞. Let xj ∈ Ω be such that

uj(xj) + |∇uj(xj)|2/(p+1) = supΩ

(uj + |∇uj |2/(p+1)

)=: Mj

and let dj := dist (xj , ∂Ω). Since Ω is compact, we may assume xj → x0 for somex0 ∈ Ω. Set κj := M

−(p−1)/2j . The sequence dj/κj is either unbounded or bounded.

In the former case we may assume dj/κj →∞, in the latter dj/κj → c ≥ 0.Case 1. Let dj/κj →∞. Set

vj(y) :=1

Mjuj(x), y :=

x− xj

κj,

and Ωj := y ∈ Rn : |y| < dj/κj. Then

vj + |∇vj |2/(p+1) ≤ vj(0) + |∇vj(0)|2/(p+1) = 1 (12.3)

and−∆vj(y) = a(κjy + xj)v

pj (y) + gj(y), y ∈ Ωj , (12.4)

where

gj(y) := κ2p/(p−1)j g

(κjy + xj , κ

−2/(p−1)j vj(y), κ(p+1)/(p−1)

j ∇vj(y))

satisfies

|gj| ≤ Cκεj , ε := min

(2(p− q), 2p− (p + 1)r

)/(p− 1). (12.5)

Interior elliptic Lp-estimates (see Appendix A) guarantee that vj are locally bound-ed in W 2,z for any z > 1 (uniformly with respect to j). Let α ∈ (0, 1), R > 0 andBR := y ∈ R

n : |y| < R. There exists z = z(α) > 1 such that W 2,z(BR) iscompactly embedded into BUC1+α(BR). Consequently, we may assume vj → v inC1+α. Passing to the limit in (12.4) and (12.3) we see that v is a positive (classical)solution of

−∆v = a(x0)vp in Rn,

which contradicts Theorem 8.1.Case 2. Let dj/κj → c ≥ 0. Let xj ∈ ∂Ω be such that dj = |xj−xj |. For any j we

can choose a local coordinate z = z(j) = (z1, z2, . . . , zn) in an ε-neighborhood Uj

of xj such that the image of the boundary ∂Ω will be contained in the hyperplanez1 = 0, xj becomes 0, xj becomes zj := (dj , 0, 0, . . . , 0), and the image of Uj willcontain the set z : |z| < ε′ for some ε′ > 0. We may assume that ε, ε′ areindependent of j and the local charts are uniformly bounded in C2. In these newcoordinates, the equation for w = wj(z) = uj(x) becomes

−∑i,k

aik(z)∂2w

∂zi∂zk−∑

i

bi(z)∂w

∂zi= a(x(z))wp + g(z), |z| < ε, z1 > 0,

w = 0, |z| < ε, z1 = 0,

⎫⎪⎬⎪⎭ (12.6)

where g(z) := g(x(z), w(z), D(z)∇zw(z)

), D = D(j) = (∂zi/∂xk)i,k, bi = bi

(j) =

∆zi, aik = aik(j) =

∑

∂zi

∂x∂zk

∂x , hence A = A(j) := (aik(j))i,k = D · tD, and the A(j)

are uniformly elliptic. Also, since ∂Ω is uniformly C2, it follows that the aik(j) are

uniformly bounded in C1 and the bi(j) in L∞. Moreover, since D(0) is a Euclidean

transformation, it follows that A(j)(0) = D(0) · tD(0) = Id. Set

vj(y, s) :=1

Mjwj(κjy + zj),

where

y ∈ Ωj :=

y :∣∣∣y − zj

κj

∣∣∣ < ε′

κj, y1 > −dj

κj

.

12. A priori bounds via the rescaling method 67

Then vj is a solution of

−∑i,k

aik(κjy + zj)∂2v

∂yi∂yk− κj

∑i

bi(κjy + zj)∂v

∂yi

= a(x(κjy + zj))vp + gj in Ωj ,

v = 0 on y ∈ ∂Ωj : y1 = −dj/κj,

where

gj(y) := κ2p/(p−1)j g

(x(κjy + zj), κ

−2/(p−1)j v(y), κ−(p+1)/(p−1)

j D(κjy + zj)∇v(y))

satisfies (12.5). Interior-boundary Lp-estimates (see Appendix A) and the boundson the coefficients aik

(j), bi(j) again yield a subsequence of vj converging to a

positive (classical) solution v of

∆v = a(x0)vp, y1 > −c,

v = 0, y1 = −c,

which contradicts Theorem 8.2.

Remarks 12.3. (i) If g is independent of the gradient variable, then it is sufficientto choose Mk := sup uk in the proof of Theorem 12.1.

(ii) Indefinite coefficients. Assume that the function a in problem (12.2)changes sign. Under suitable assumptions on a, g and p one can still use the methodof [241] in order to get a priori bounds for positive solutions (see [74], [19] and [167],for example). In addition to the limiting problems in the proof of Theorem 12.1one has to deal with problems of the form

−∆u = h(y)up, y ∈ Rn,

where typically h(y) = |y1|αy1 for some α ≥ 0. In some cases, a combination ofthe above approach with other arguments (moving planes, energy, . . . ) yields thea priori bounds, see [124], [453], [237] and the references therein. Of course, if theproblem has variational structure, then the existence of nontrivial solutions canoften be proved by variational or dynamical methods, see [8], [75], [7], [257], [121],[3] and the references therein.

(iii) The rescaling method is sometimes referred to as the “blow-up method”,because one performs a zoom of the microscopic scales of the solution. Here we shallnot use this terminology, in order to avoid confusion with the blow-up phenomenonin the parabolic problem.

13. A priori bounds via moving planes andPohozaev’s identity

In this section we describe the method of a priori estimates of solutions of (2.1) dueto [183]. Similarly as in the preceding section, the growth condition for functionf will be optimal. The advantage of this method consists in the fact that it doesneither require precise asymptotic behavior of f for u large nor Liouville-typetheorems. On the other hand, the symmetry of the Laplace operator plays animportant role, f cannot depend on ∇u in a general way and we also have toassume that either Ω is convex or f satisfies a restrictive monotonicity condition,see (13.3) below. The assumptions for a general function f = f(x, u) are rathercomplicated (see [183, Remark 1.5]) and therefore we restrict ourselves to the casef = f(u). Hence, we shall study positive solutions of the problem

−∆u = f(u), x ∈ Ω,

u = 0, x ∈ ∂Ω.

(13.1)

Theorem 13.1. Assume n ≥ 2 and Ω bounded. Let f : R+ → R be locallyLipschitz continuous and assume

lim infu→∞

f(u)u

> λ1, limu→∞

f(u)uσ

= 0,

where σ = pS if n ≥ 3, σ < ∞ is arbitrary if n = 2. Let one of the followingassumptions be satisfied:

(i) Ω is convex and

lim supu→∞

uf(u)− θF (u)u2f(u)κ

≤ 0, θ ∈ [0, 2∗), (13.2)

where κ = 2/n.(ii) Condition (13.2) is satisfied with κ = 2/n and, in the case n ≥ 3,

the function u → f(u)u−pS is nonincreasing on (0,∞). (13.3)

(iii) Condition (13.2) is satisfied with κ = 2/(n+1), n ≥ 3, ∂Ω = Γ1∪Γ2, whereΓ1, Γ2 are closed and satisfy

(1) at every point of Γ1, all sectional curvatures of Γ1 are bounded awayfrom 0 by a positive constant a;

(2) there exists x0 ∈ Rn such that (x − x0, ν(x)) ≤ 0 for all x ∈ Γ2.

Then there exists C > 0 such that ‖u‖∞ < C for any positive classical solutionu of (13.1).

In view of the proof we set some notation. For each ε > 0, let

Ωε := z ∈ Ω : δ(z) < ε.

13. A priori bounds via moving planes and Pohozaev’s identity 69

For y ∈ ∂Ω and λ > 0, we define

T (y, λ) := x ∈ Rn : (y − x, ν(y)) = λ,

Σ(y, λ) := x ∈ Ω : (y − x, ν(y)) ≤ λ,

we denote by R(y, λ) the reflection with respect to the hyperplane T (y, λ) and weset Σ′(y, λ) := R(y, λ)Σ(y, λ). We need the following lemma.

Lemma 13.2. Assume Ω bounded and convex, λ0 > 0, and 0 ≤ u ∈ C(Ω)∩C1(Ω).Assume that

(∇u(x), ν(y)) ≤ 0, y ∈ ∂Ω, x ∈ Σ(y, λ0). (13.4)

ThensupΩε

u ≤ C

∫Ω

uϕ1 dx,

where ε, C > 0 depend only on Ω and λ0.

Proof. Let us first recall that

ν(∂Ω) = Sn−1. (13.5)

This follows from a standard degree argument. We give the proof for completeness.Assume without loss of generality that 0 ∈ Ω and select ν, a continuous extensionof ν to Ω. The homotopy H1(t, x) := tν(x) + (1 − t)x has no zero on ∂Ω, since(x, ν(x)) ≥ 0 on ∂Ω due to the convexity of Ω. Therefore d(ν, 0, Ω) = d(id, 0, Ω) =1, where d denotes the Brouwer degree. Assume for contradiction that η ∈ ν(∂Ω)for some η ∈ Sn−1. Then the homotopy H2(t, x) = tν(x) − (1 − t)η has no zeroon ∂Ω. Consequently d(ν, 0, Ω) = d(−η, 0, Ω) = 0, a contradiction which proves(13.5).

Next, by decreasing λ0 if necessary, we may assume that

y − λν(y) ∈ Rn : λ ∈ (0, λ0] ⊂ Ω, y ∈ ∂Ω. (13.6)

Let ε ∈ (0, λ0/4], x ∈ Ωε, and let x ∈ ∂Ω satisfy |x − x| = δ(x). Notice that x isuniquely determined and (x − x)/|x − x| = ν(x) if ε is small. Let α ∈ (0, 1) andlet η ∈ Sn−1 be such that (η, ν(x)) ≥ α. Using the fact that Ω is contained in thehalf-space z ∈ R

n : (z−x, ν(x)) ≤ |x−x| (due to the convexity of Ω), we obtain

(y(η)−x, η) ≤ (y(η)−x, ν(x))+|y(η)−x||η−ν(x)| ≤ ε+diam(Ω)√

2(1− α) ≤ λ0/2,

provided α is close to 1 and ε is small enough, say 1−α+ε < ε0 = ε0(Ω, λ0). Thisalong with (13.6) implies

x− λη ∈ Rn : λ ∈ [0, λ0] ⊂ Σ(y(η), λ0).

It then follows from (13.4) that [0, ε] λ → u(x − λη) is nondecreasing for anyη ∈ Sn−1 satisfying (η, ν(x)) ≥ α. This property guarantees the existence ofγ = γ(Ω, λ0) > 0 such that

for all x ∈ Ωε there exists a measurable set Ix ⊂ Ω \Ωε

satisfying meas Ix ≥ γ and u(ξ) ≥ u(x) for all ξ ∈ Ix.

(13.7)

Indeed (decreasing the value of ε if necessary), it is sufficient to take a conicalpiece

Ix = Ωcε ∩ x− λη : η ∈ Sn−1, (η, ν(x)) ≥ α, λ ∈ [0, λ0].

Since ϕ1 ≥ Cε on Ω \ Ωε for some Cε > 0, we deduce from (13.7) that

Cεγu(x) ≤ Cε

∫Ix

u(ξ) dξ ≤∫

Ix

u(ξ)ϕ1(ξ) dξ ≤∫

Ω

u(ξ)ϕ1(ξ) dξ

and the lemma is proved.

Proof of Theorem 13.1. First assume (i). The proof will consist of the followingfour steps:

1.∫Ω

uδ dx ≤ C,∫Ω|f(u)|δ dx ≤ C, where δ(x) = dist (x, ∂Ω),

2. u + |∇u| ≤ C in a neighborhood of ∂Ω,3. ‖∇u‖2 ≤ C,4. ‖u‖∞ ≤ C.

Step 1. This step is almost the same as Step 1 in the proof of Theorem 10.1 andwe leave the detailed proof to the reader.

Step 2. Since Ω is convex and smooth, we can find λ0, c0 > 0 such that

Σ′(y, λ) ⊂ Ω, λ ≤ λ0 and (ν(x), ν(y)) > c0, x ∈ ∂Σ(y, λ0) ∩ ∂Ω.

We shall now apply the moving planes method (cf. [239], [183]) to show that

u(R(y, λ)x) ≥ u(x), y ∈ ∂Ω, x ∈ Σ(y, λ), λ ≤ λ0. (13.8)

Without loss of generality, we may assume that y = 0 and that ν(0) = −e1 (inparticular, Ω lies entirely in the upper half-space x1 > 0). For each x = (x1, x

′),we denote xλ := R(0, λ)x = (2λ − x1, x

′), Σλ := Σ(0, λ) = Ω ∩ x1 < λ, andΣ′

λ := Σ′(0, λ). Define

wλ(x) = u(xλ)− u(x), for x ∈ Σλ, 0 < λ ≤ λ0,

and set

E :=µ ∈ (0, λ0] : wλ(x) ≥ 0 for all x ∈ Σλ and λ ∈ (0, µ)

.

Since ∂u∂x1

(0) > 0 by Hopf’s lemma, we have λ ∈ E for λ > 0 small. Assume forcontradiction that λ := supE < λ0. We have

wλ ≥ 0, for all x ∈ Σλ and λ ∈ (0, λ], (13.9)

and there exists a sequence λi → λ, with λ < λi < λ0, such that minΣλi

wλi < 0.

Since wλ = 0 on x1 = λ ∩Ω and

wλ > 0 on x1 < λ ∩ ∂Ω, for all λ < λ0, (13.10)

it follows that this minimum is attained at a point qi ∈ Σλi . Therefore ∇wλi (qi) =0. On the other hand, since ∂u

∂x1= (e1 · ν) ∂u

∂ν ≥ c > 0 on x1 ≤ λ0 ∩ ∂Ω and

wλ(x) = u(2λ− x1, x′)− u(x1, x

′) = 2(λ− x1)∂u

∂x1(ξ(x)),

with |ξ(x) − x| ≤ 2(λ− x1), we see that wλ(x) ≥ 0 for x in an ε-neighborhood ofx1 = λ ∩ ∂Ω, with ε > 0 independent of λ ∈ (0, λ0]. Therefore, we may assumethat qi → q ∈ Σλ, q /∈ x1 = λ ∩ ∂Ω, and by continuity we get

wλ(q) = 0 and ∇wλ(q) = 0. (13.11)

But (13.9) implies

−∆wλ(x) = f(u(xλ)

)− f(u(x)) ≥ −cwλ(x) and wλ(x) ≥ 0, x ∈ Σλ,

for some constant c > 0 (depending on u). By Hopf’s lemma (cf. Proposition 52.1and Remark 52.2), this along with (13.11) implies wλ = 0 in Σλ, contradicting(13.10). Consequently, λ = λ0, which proves (13.8). This guarantees that u satisfies(13.4). By Lemma 13.2 and Step 1, we deduce that u ≤ C on Ωε for some ε, C > 0depending only on Ω. Now the bound for∇u in Ωε/2 follows from interior-boundaryelliptic Lp-estimates (see Appendix A) and the embedding W 2,p → C1 for p > n.In particular, we have shown that∣∣∣∂u

∂ν

∣∣∣ ≤ C, x ∈ ∂Ω. (13.12)

Step 3. Notice that Steps 1 and 2 imply

‖f(u)‖1 ≤ C. (13.13)

First consider the case n ≥ 3. The Holder and Sobolev inequalities and (13.13)guarantee ∫

Ω

u2|f(u)|2/n dx ≤ ‖u‖22∗‖f(u)‖2/n1 ≤ C‖∇u‖22.

Pohozaev’s identity (5.1) and (13.12) yield∣∣∣ ∫Ω

|∇u|2 dx− 2∗∫

Ω

F (u) dx∣∣∣ ≤ C.

Since∫Ω |∇u|2 dx =

∫Ω uf(u)dx, the last two estimates and (13.2) imply

2∗∫

Ω

F (u) dx ≤∫

Ω

uf(u)dx + C ≤ θ

∫Ω

F (u) dx + ε

∫Ω

u2|f(u)|2/n dx + Cε

≤ (θ + εC)∫

Ω

F (u) dx + Cε.

Choosing ε < (2∗ − θ)/C we obtain∫Ω F (u) dx ≤ C, hence ‖∇u‖2 ≤ C.

Next let n = 2. Set γ := 1− 1/σ. Given ε > 0, the assumption limu→∞ f(u)/uσ

= 0 guarantees the existence of Cε > 0 such that

uf(u) ≤ εu2f(u)γ + Cε.

Similarly as above we obtain

‖∇u‖22 =∫

Ω

uf(u)dx ≤ ε

∫Ω

u2|f(u)|γ dx + Cε

≤ ε‖u‖22/(1−γ)‖f(u)‖γ1 ≤ εC‖∇u‖22 + Cε,

which proves the assertion.Step 4. If

f(u) ≤ C(1 + up) for some p < pS (13.14)

(which is always true if n = 2), then one can use standard bootstrap estimatesbased on Lq-estimates (see Appendix A) to show that the W 1,2-bound from Step 3guarantees an L∞-bound. If n ≥ 3 and (13.14) is not true, then we use the followingestimates (see [96] and cf. the proof of Proposition 3.3).

Let p > 1, ap := (p + 1)2/4 and q := (p + 1)n/(n− 2). Then(∫Ω

uq dx)(n−2)/n

=∥∥u(p+1)/2

∥∥2

2∗ ≤ C

∫Ω

∣∣∇u(p+1)/2∣∣2 dx = Cap

∫Ω

|∇u|2up−1 dx

= Cap

p

∫Ω

f(u)up dx ≤ ε

∫Ω

up+σ dx + Cε,

where σ = (n + 2)/(n− 2). Next Holder’s inequality and Step 3 yield∫Ω

up+σ dx =∫

Ω

uq(n−2)/n+4/(n−2) dx ≤(∫

Ω

uq dx)(n−2)/n(∫

Ω

u2∗dx)2/n

≤ C(∫

Ω

uq dx)(n−2)/n

.

These estimates imply ‖u‖q ≤ C, hence ‖f(u)‖q/σ ≤ C. Since q can be madearbitrarily large, the Lp-estimates (see Appendix A) conclude the proof in case(i).

Next consider assumption (ii). Instead of Ω being convex we now assume (13.3).Since the convexity assumption was used only in the proof of Step 2, it is sufficientto modify the proof of this step. Choose x0 ∈ ∂Ω. Then there exists a ball Br ⊂R

n\Ω of radius r such that x0 ∈ ∂Br. The radius r can be chosen independent of x0

and, without loss of generality, we may assume r = 1. Choose a coordinate systemsuch that Br is centered at the origin and x0 = (1, 0, . . . , 0). Set y = J(x) := x/|x|2and w(y) = |x|n−2u(x). Then

−∆w(y) = g(y, w) in O := J(Ω),

where g(y, w) := f(|y|n−2w)/|y|n+2 is nonincreasing in y due to (13.3). SinceO ⊂ Br is smooth and x0 ∈ ∂O ∩ ∂Br we can use the moving planes methodin order to get the existence of εx0 , γx0 > 0 with the following property: for anyy ∈ O, |y−x0| < εx0 , there exists a set Ky ⊂ z ∈ O : dist (z, ∂O) > εx0 satisfyingmeasKy ≥ γx0 and w(ξ) ≥ w(y) for all ξ ∈ Ky. Going back to the original variablesand using the compactness of ∂Ω we get the existence of ε, γ, c > 0 such that (13.7)is true, with u(ξ) ≥ u(x) replaced by u(ξ) ≥ cu(x). The rest of the proof of Step 2is the same as in case (i).

Finally consider case (iii). Then Steps 1 and 4 can be proved in the same way asin case (i). Repeating the arguments in the proof of Step 2 of case (i) we obtain auniform bound for u and |∇u| in a neighborhood of Γ1. Without loss of generalitywe may assume x0 = 0, hence x·ν(x) ≤ 0 for all x ∈ Γ2. These facts and Pohozaev’sidentity (5.1) imply

2∗∫

Ω

F (u) dx−∫

Ω

uf(u)dx ≤ C. (13.15)

Next using Lemma 50.4 with τ := 1/(n + 1) and q := 2(n + 1)/(n− 1), Step 1 andHolder’s inequality, we obtain∫

Ω

uf(u)dx = ‖∇u‖22 ≥ c1

∥∥∥ u

δτ

∥∥∥2

q≥ c2

∥∥∥ u

δτ

∥∥∥2

q‖f(u)δ‖1−2/q

1

≥ c2

∫Ω

u2

δ2τ

(|f(u)|δ

)1−2/qdx = c2

∫Ω

u2|f(u)|2/(n+1) dx.

Now (13.2) with κ = 2/(n + 1), (13.15) and the last estimate imply∫Ω

uf(u)dx ≤ θ

∫Ω

F (u) dx + ε

∫Ω

u2|f(u)|2/(n+1) dx + Cε

≤ (θ/2∗ + εC)∫

Ω

uf(u)dx + Cε

and the choice of ε small enough concludes the proof.

The following corollary can be proved in the same way as Corollary 10.3.

Corollary 13.3. Let f : R+ → R+ satisfy the assumptions in Theorem 13.1and lim supu→0+ f(u)/u < λ1. Then problem (2.1) possesses at least one positiveclassical solution.

Remark 13.4. If one is interested only in the existence of positive solutions of(2.1) without knowing their a priori bounds, then the technical assumption (13.2)can be omitted, see [183]. The proof is based on an approximation of the functionf , on the mountain pass theorem (including uniform bounds for the energy ofapproximating solutions) and Pohozaev’s identity.

Chapter II

Model Parabolic Problems

14. Introduction

In Chapter II, we mainly consider semilinear parabolic problems of the form

ut −∆u = f(u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (14.1)

where f is a C1-function with a superlinear growth. For simplicity, we formulatemost of our assertions for the model case f(u) = |u|p−1u with p > 1, but themethods of our proofs can be applied to more general parabolic problems (notnecessarily of the form (14.1)). Some of possible generalizations and modificationswill be mentioned as remarks, other can be found in the subsequent chapters.

15. Well-posedness in Lebesgue spaces

Definition 15.1. Given a Banach space X of functions defined in Ω, u0 ∈ X andT ∈ (0,∞], we say that the function u ∈ C([0, T ), X) is a solution (more precisely,a classical X-solution) of (14.1) in [0, T ) if u ∈ C2,1(Ω× (0, T ))∩C(Ω× (0, T )),u(0) = u0 and u is a classical solution of (14.1) for t ∈ (0, T ). If Ω is unbounded,then we also require u ∈ L∞

loc((0, T ), L∞(Ω)).If X = L∞(Ω), then, instead of the condition u ∈ C([0, T ), X), we require

u ∈ C((0, T ), X) and ‖u(t)− e−tAu0‖∞ → 0 as t→ 0, where e−tA is the Dirichletheat semigroup in Ω (cf. Appendix B).

We say that (14.1) is well-posed in X if, given u0 ∈ X , there exist T > 0 anda unique classical X-solution of (14.1) in [0, T ].

It is well known that (14.1) is well-posed in X = W 1,q0 (Ω) for any q > n if Ω is

bounded, or in X = L∞(Ω) for any Ω (see Example 51.9 and Remark 51.11). Inthis section we study the well-posedness of the model problem

ut −∆u = |u|p−1u, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (15.1)

76 II. Model Parabolic Problems

in the Lebesgue spaces Lq(Ω), 1 ≤ q < ∞, since this well-posedness will play acrucial role in many subsequent sections. The following two results show that theexponent

qc := n(p− 1)/2

is critical for the well-posedness. The existence/nonexistence part of these resultsis due to [528], [529] (where uniqueness was proved in a more restrictive classof solutions). The uniqueness and nonuniqueness parts were proved in [93] and[53], [425], respectively. An alternative proof of the existence-uniqueness part ofTheorem 15.2 based on interpolation and extrapolation spaces can be found inAppendix E (see Theorem 51.25 and Example 51.27).

In what follows we write shortly Lq-solution instead of Lq(Ω)-solution.

Theorem 15.2. Let p > 1, u0 ∈ Lq(Ω), 1 ≤ q < ∞, q > qc. Then there existsT = T (‖u0‖q) > 0 such that problem (15.1) possesses a unique classical Lq-solutionin [0, T ) and the following smoothing estimate is true:

‖u(t)‖r ≤ C‖u0‖qt−αr , αr :=

n

2

(1q− 1

r

), (15.2)

for all t ∈ (0, T ) and r ∈ [q,∞], with C = C(n, p, q) > 0. In addition, u ≥ 0provided u0 ≥ 0.

Theorem 15.3. Let p > 1 + 2/n and 1 ≤ q < qc.

(i) There exists a nonnegative function u0 ∈ Lq(Ω), such that (15.1) does not admitany nonnegative classical Lq-solution in [0, T ) for any T > 0.

(ii) Assume p < pS, Ω = BR, and let u0 ∈ L∞(Ω), u0 ≥ 0, be radial nonincreasing.Then there exists a time T > 0 such that (15.1) possesses infinitely many positiveradial nonincreasing classical Lq-solutions in [0, T ).

Remarks 15.4. (i) The critical case. It was also proved in [529], [93] thatwhen u0 ∈ Lq(Ω) and q = qc > 1, then there exists T = T (u0) > 0 such that(15.1) possesses a unique classical Lq-solution in [0, T ) (see Example 51.27 andcf. Remark 20.24(i)). The same arguments as in Remark 51.26(vi) guarantee thatthis solution satisfies (15.2) in (0, T ). In addition, it is nonnegative if u0 ≥ 0.

Unlike in the case q > qc, T cannot be chosen uniform for all u0 lying ina bounded subset of Lq(Ω). Indeed assume without loss of generality that Ω ⊃B(0, 1) and choose 0 ≤ u0 ∈ Lq ∩ L∞(Ω) such that T0 := Tmax(u0) < ∞ (seeSection 16 for the definition of the maximal existence time Tmax(u0) and Section 17for the existence of such solution). For each j ≥ 1, set ωj = B(0, 1/j) and define

u0,j(x) :=

j2/(p−1)u0(jx), x ∈ ωj ,

0, x ∈ Ω \ ωj .(15.3)

15. Well-posedness in Lebesgue spaces 77

By direct computation, we see that uj(x, t) := j2/(p−1)u(jx, j2t) solves (15.1) inωj × (0, j−2T0) with initial data u0,j |ωj

. Let uj be the solution of problem (15.1)(in Ω) with initial data u0,j. Since uj ≥ 0 on ∂ωj, it follows from the comparisonprinciple that uj ≥ uj in ωj as long as uj exists. Consequently, Tmax(u0,j) ≤j−2T0 → 0, as j → ∞, while ‖u0,j‖qc = ‖u0‖qc due to qc = n(p − 1)/2. SeeRemark 22.10(iii), Remark 27.8(g) and [36] for further results in that direction.

If q = qc = 1 (i.e. q = 1, p = 1 + 2/n), then there exists a positive functionu0 ∈ L1(Ω) for which (15.1) does not possess any nonnegative classical L1-solutionin [0, T ) for any T > 0; see [93, Theorem 11]) and see also [116] for a similar examplewith the weaker notion of integral solution.

(ii) Nonuniqueness in Rn. Assume Ω = R

n, u0 = 0, 1 + 2/n 0 and which is a globalclassical Lq-solution of (15.1) for any q < qc (and a W 1,q-solution for any q <n(p−1)/(p+1), see [269]). Moreover, there exist infinitely many nontrivial functionswhich are global classical Lq-solutions of (15.1) for any q < qc (see [532]). All thesesolutions are (forward) self-similar, that is

u(x, t) = λ2/(p−1)u(λx, λ2t), λ > 0.

Such solutions can be found in the form u(x, t) = t−1/(p−1)w(x/√

t), where w =w(y) solves the problem

∆w +y

2· ∇w +

1p− 1

w + |w|p−1w = 0, y ∈ Rn. (15.4)

In [269] and [532], radial positive and infinitely many radial nontrivial solutionsof (15.4) (with a rapid decay at infinity) were found by ODE techniques (see also[533]). Variational methods for solving (15.4) were used in [174].

(iii) Uniqueness and nonuniqueness in the class of mild solutions. Ifu is a classical Lq-solution of (14.1) in [0, T ), then it satisfies the variation-of-constants formula

u(t) = e−(t−τ)Au(τ) +∫ t

τ

e−(t−s)Af(u(s)) ds, 0 < τ < t < T. (15.5)

Indeed, applying the operator e−(t−s)A to the equation ut(s) + Au(s) = f(u(s)),integrating in s ∈ (τ, t) and using d

ds(e−(t−s)Au(s)) = e−(t−s)A(ut(s) + Au(s)) weobtain (15.5).

Any function u ∈ C([0, T ), Lq(Ω)) satisfying f(u) ∈ L1loc((0, T ), L1 + L∞(Ω)),

u(0) = u0 and (15.5) is called a mild Lq-solution of (15.1). (If q = ∞, then wemodify this definition in the same way as in the case of classical solutions.)

Now assume q ≥ p and let u be a mild Lq-solution of (15.1). Then we can passto the limit in (15.5) as τ → 0 to get

u(t) = e−tAu0 +∫ t

0

e−(t−s)A|u(s)|p−1u(s) ds. (15.6)

On the other hand, any solution of (15.6) in C([0, T ), Lq(Ω)

)is obviously a mild

Lq-solution. If, in addition, q ≥ qc (and q > p if q = qc), then each mild Lq-solutionis a classical Lq-solution so that the uniqueness in Theorem 15.2 and (i) holds inthe class of mild Lq-solutions (see [93], [536]). This is not true for the limiting caseq = qc = p = n/(n− 2). In fact, if Ω is the unit ball and q = p = n/(n− 2), thenthere exists a singular stationary solution us ∈ Lq(Ω) \ C(Ω) of (15.1) (see [394]and cf. Remark 3.6(ii)). The function u(t) := us is a mild Lq-solution of (15.1)with u0 := us which is not classical for t > 0. On the other hand, (i) guarantees theexistence of a classical Lq-solution. A similar example for Ω = R

n was constructedin [511].

(iv) Integral solutions. Consider problem (14.1) with f nonnegative and u0 ≥0. We say that u is an integral solution of (14.1) in [0, T ) if u : Ω×[0, T )→ [0,∞]is measurable, finite a.e. and

u(x, t) =∫

Ω

G(x, y, t)u0(y) dy +∫ t

0

∫Ω

G(x, y, t− s)f(u(y, s)) dy ds (15.7)

for a.e. (x, t) ∈ QT , where G is the Dirichlet heat kernel in Ω (cf. Appendix B). Ifu0 ∈ Lq(Ω) is nonnegative and u is a mild Lq-solution of (14.1), then u is also anintegral solution of (14.1). In fact, since

e−tAw(x) =∫

Ω

G(x, y, t)w(y) dy,

the functions u, f are nonnegative and u : [0, T )→ Lq(Ω) is continuous, it is easyto pass to the limit in (15.5) as τ → 0 in order to obtain (15.7). Let us mentionthat the nonexistence statement in Theorem 15.3 is true in the class of integralsolutions.

(v) Weak solutions. Assume that Ω is bounded and u0 ∈ L1δ(Ω). A func-

tion u ∈ C([0, T ), L1δ(Ω)) is called a weak solution (more precisely weak L1

δ-solution) of (14.1) in [0, T ) if the functions u, δf(u) belong to L1

loc((0, T ), L1(Ω)),u(0) = u0 and ∫ t

τ

∫Ω

f(u)ϕ = −∫ t

τ

∫Ω

u(ϕt + ∆ϕ)−∫

Ω

u(τ)ϕ(τ)

for any 0 < τ < t < T and any ϕ ∈ C2(Ω × [τ, t]) such that ϕ = 0 on ∂Ω× [τ, t]and ϕ(t) = 0. One can prove that any mild Lq-solution (hence any classical Lq-solution) is a weak L1

δ-solution for any q ≥ 1 (see Corollary 48.11) and that thelinear problem

ut −∆u = f, x ∈ Ω, t ∈ [0, T ),

u = 0, x ∈ ∂Ω, t ∈ [0, T ),

u(x, 0) = u0(x), x ∈ Ω(15.8)

possesses a unique weak L1δ-solution in [0, T ) for any f ∈ L1

loc([0, T ), L1δ(Ω)) and

u0 ∈ L1δ(Ω) (see Proposition 48.9). It is also easy to see that the notions of integral

solution and weak L1δ-solution coincide if Ω is bounded, f is nonnegative and we

consider nonnegative, locally integrable solutions only (see Corollary 48.10). Noticethat uniqueness of weak solutions need not be true for the nonlinear problem (see(iii)).

(vi) Initial traces. In view of Theorems 15.2 and 15.3, it is a natural questionto ask what should be the most general admissible initial data for local existencein problem (15.1). This question can be formulated as a problem of initial tracesand has been studied in [56], [28]. In the case Ω = R

n, it is known that for anynonnegative classical solution u of

ut −∆u = up, x ∈ Rn, 0 < t < T (15.9)

with p > 1, there exists a unique nonnegative Radon measure µ such that

u(t)→ µ in the sense of measures, as t→ 0. (15.10)

The measure µ is called the initial trace of u. If, moreover, p < 1 + 2/n, then µis uniformly locally finite, i.e.:

supx∈Rn

∫B(x,1)

dµ < ∞, (15.11)

and the result is optimal. Namely, if p < 1 + 2/n and µ is any nonnegative Radonmeasure verifying (15.11), then there exists a nonnegative classical solution of(15.9) which satisfies (15.10), and it is unique in a suitable class. Actually theseresults remain valid for properly defined weak solutions. On the other hand, in therange p ≥ 1 + 2/n, the initial trace of a given solution u has to satisfy conditionsstronger than (15.11) (in particular the Dirac measure µ = δ0 is not admissible),but the necessary and sufficient condition on initial traces seems to be unknown.See also Theorem 15.11 and Remark 15.12 below for related results.

Remark 15.5. Independence of the local solution with respect to q. Ifu0 ∈ Lq1 ∩ Lq2(Ω) for some 1 ≤ q1, q2 ≤ ∞, with q1, q2 > qc or q1, q2 ≥ qc > 1,then the corresponding solutions ui on [0, T i), given by Theorem 15.2 (or Re-mark 15.4(i)), coincide for t < min(T 1, T 2). This is a consequence of the followinggeneral argument.

By decreasing one of the Ti’s, we may assume T1 = T2. The solution ui isobtained as the unique fixed point of a contraction Φi

u0: X i → X i, where X i is a

complete metric space (of functions of t ∈ [0, T1)). For u0 as above, Φ1u0

coincideswith Φ2

u0on X := X1 ∩X2, and it is a contraction on the complete metric space

X (with norm ‖ · ‖X = max(‖ · ‖X1 , ‖ · ‖X2). It thus has a unique fixed point u. Byuniqueness in each X i, we immediately deduce that u1 = u = u2.

Proof of Theorem 15.2. It is divided into several steps.Step 1. Fixed-point argument. To handle the singularity of the initial data, the

idea is to introduce a Banach space of functions with a temporal weight which hasa suitable decay as t → 0. We may assume that ‖u0‖q > 0. Let T > 0 be smalland consider the Banach space

YT := u ∈ L∞loc

((0, T ), Lpq(Ω)

): ‖u‖YT < ∞, ‖u‖YT := sup

0<t<Ttα‖u(t)‖pq,

where α := n(p − 1)/2pq < 1/p < 1. Choose M > ‖u0‖q and let BM = BM,T

denote the closed ball in YT with center 0 and radius M . We will use the Banachfixed point theorem for the mapping Φu0 : BM → BM , where

Φu0(u)(t) := e−tAu0 +∫ t

0

e−(t−s)A|u(s)|p−1u(s) ds. (15.12)

Using the Lp-Lq-estimates (see Proposition 48.4) we obtain for any u, v ∈ BM

and v0 ∈ Lq(Ω),

tα‖Φu0(u)(t)− Φv0(v)(t)‖pq

≤ tα‖e−tA(u0 − v0)‖pq + tα∫ t

0

‖e−(t−s)A(|u(s)|p−1u(s)− |v(s)|p−1v(s)

)‖pq ds

≤ (4π)−α‖u0 − v0‖q + tα∫ t

0

[4π(t− s)]−α‖(|u(s)|p−1u(s)− |v(s)|p−1v(s)

)‖q ds

≤ (4π)−α‖u0 − v0‖q

+ C′(p)tα∫ t

0

(t− s)−α(‖u(s)‖p−1

pq + ‖v(s)‖p−1pq

)‖u(s)− v(s)‖pq ds

≤ (4π)−α‖u0 − v0‖q + C(p)Mp−1tα∫ t

0

(t− s)−αs−(p−1)α‖u(s)− v(s)‖pq ds.

(15.13)In particular, choosing v0 = 0 and v = 0 in (15.13) we have

‖Φu0(u)‖YT ≤ (4π)−α‖u0‖q + sup0<t<T

C(p)Mp−1tα∫ t

0

(t− s)−αs−pα ds ‖u‖YT

≤ (4π)−α‖u0‖q + C(p, α)Mp−1T 1−pα‖u‖YT .

Let T0 = T0(M, n, p, q) > 0 be such that

C(p, α)Mp−1T 1−pα0 < min

(1− (4π)−α, 1/2

)and C(p)Mp−1T 1−α

0 < 1/2.(15.14)

Then the above estimate implies

‖Φu0(u)‖YT < (4π)−αM + (1− (4π)−α)M = M for any T ≤ T0, (15.15)

hence Φu0 maps BM into BM for T ≤ T0. Choosing v0 = u0 in (15.13) we obtain

‖Φu0(u)− Φu0(v)‖YT ≤ C(p, α)Mp−1T 1−pα‖u− v‖YT ≤12‖u− v‖YT

for any T ≤ T0. Consequently, Φu0 is a contraction in BM and it possesses a uniquefixed point u in this set.

Note for further reference that in fact, for any T ≤ T0,

u is the only solution of Φu0(u) = u in YT . (15.16)

Indeed, given any two solutions, both belong to BM ′,T ′0

for some large M ′ and smallT ′

0 satisfying (15.14). Therefore they coincide for small t > 0, hence on (0, T ) byan obvious continuation argument.

Step 2. Regularity. The function u satisfies |u|p−1u ∈ L1((0, T ), Lq(Ω)

)hence

u = Φu0(u) ∈ C([0, T ], Lq(Ω)

). Choose ε > 0 small and set κ1 := pq. Then

u ∈ L∞([ε, T ], Lκ1(Ω))

and

u(t + ε) = e−tAu(ε) +∫ t

0

e−(t−s)A|u(s + ε)|p−1u(s + ε) ds. (15.17)

Choose κ2 > κ1 such that β1 := n2

(pκ1− 1

κ2

)< 1 and set β2 := n

2

(1κ1− 1

κ2

). Using

(15.17) and the Lp-Lq-estimates we get

‖u(t + ε)‖κ2 ≤ t−β2‖u(ε)‖κ1 +∫ t

0

(t− s)−β1‖u(s + ε)‖pκ1

ds ≤ C(ε)

for t ∈ [ε, T − ε]. Hence u ∈ L∞([2ε, T ], Lκ2(Ω))

and an obvious bootstrap argu-ment shows u ∈ L∞

loc

((0, T ], L∞(Ω)

). Now standard existence and regularity results

for linear parabolic equations (see Appendix B) guarantee that u is a classical so-lution for t > 0, hence a classical Lq-solution. Let us explain this in more detailin the case of bounded domains; in the general case one can use smooth cut-offfunctions and use localized versions of the regularity statements in Appendix B.

Fix δ > 0 small and let ψ : R → [0, 1] be a smooth function satisfying ψ(t) = 0for t ≤ δ and ψ(t) = 1 for t ≥ 2δ. Since u is a mild solution, it is also a weak(L1

δ-) solution (see Corollary 48.11). Consequently, ψu is a weak solution of thelinear problem (15.8) with f := ψtu + ψ|u|p−1u ∈ L∞(Q), where Q := QT . NowTheorem 48.1(iii) guarantees that this linear problem has a strong solution v ∈W 2,1;q(Q) for any q ∈ (1,∞). This strong solution is obviously a weak solutionand the uniqueness of weak solutions (see Proposition 48.9) guarantees ψu = v,consequently u ∈ W 2,1;q(Ω × (2δ, T )). Now fixing q > n + 2 we see that f(u) is

Holder continuous in Ω× (2δ, T ). Next consider the function ψ(t−2δ)u(t) and useTheorem 48.2(ii) to see that u is a classical solution for t > 4δ.

Step 3. Continuous dependence. Let us denote by U(t)u0 the solution u(t) con-structed above. The existence proof shows that U(·)v0 is defined and belongs toBM,T for any v0 satisfying ‖v0‖q < M and any T ≤ T0. In addition, (15.13)guarantees

‖U(·)u0 − U(·)v0‖YT ≤ ‖u0 − v0‖q + C(p, α)Mp−1T 1−pα‖U(·)u0 − U(·)v0‖YT ,

hence the choice of T0 implies

‖U(·)u0 − U(·)v0‖YT ≤ 2‖u0 − v0‖q. (15.18)

It follows that

‖U(t)u0 − U(t)v0‖q

≤ ‖u0 − v0‖q +∫ t

0

‖|U(s)u0|p−1U(s)u0 − |U(s)v0|p−1U(s)v0‖q ds

≤ ‖u0 − v0‖q + C(p)Mp−1T 1−α0 ‖U(·)u0 − V (·)v0‖YT

≤ 2‖u0 − v0‖q

(15.19)whenever t ≤ T0. Consequently, the map Lq(Ω) → Lq(Ω) : v0 → U(t)v0 is Lipschitzcontinuous in a neighborhood of u0.

Step 4. Uniqueness. Let v be a classical Lq-solution of (15.1) in an interval[0, T1), that is v ∈ C

([0, T1), Lq(Ω)

)∩L∞

loc((0, T1), L∞(Ω)), v(0) = u0 and v is aclassical solution of (15.1) for t ∈ (0, T1). Due to the uniqueness property (15.16),it is sufficient to show that v(t) = U(t)u0 for small t. Decreasing T1 if necessary wemay thus assume that T1 ≤ T0 and ‖v(s)‖q < M for all s ∈ [0, T1). Let T = T1/2.For each τ ∈ (0, T ), since vτ := v(· + τ) ∈ YT and vτ satisfies (15.6), property(15.16) implies

v(t + τ) = U(t)v(τ) for all t ∈ (0, T ).

Passing to the limit as τ → 0 and using (15.19), we obtain v(t) = U(t)u0 for allt ∈ (0, T ), hence the solution u is unique.

Step 5. Smoothing estimate. Fix M = 2‖u0‖q and notice that T0 = T0(‖u0‖q)(provided we suppress the dependence of T0 on n, p, q). Choose r ≥ q. If r = qor r = pq, then (15.2) follows from (15.19) (with v0 = 0) or (15.15), respectively.Assume that

‖u(t)‖m ≤ C‖u0‖qt−αm (15.20)

for some m ≥ max(p, q), where αm is defined in (15.2). We shall prove that wemay increase the value of m in this estimate (by enlarging C if necessary) in such

a way that we can reach the value m = ∞ in a finite number of iterations. Then(15.2) follows for any r ∈ [q,∞] from the interpolation inequality

‖u(t)‖r ≤ ‖u(t)‖q/rq ‖u(t)‖1−q/r

∞ .

Similarly as above we obtain

‖u(t)‖r ≤ ‖e−tA/2u(t/2)‖r +∫ t

t/2

(t− s)−(n/2)(p/m−1/r)‖u(s)‖pm ds

≤ t−(n/2)(1/m−1/r)‖u(t/2)‖m

+ Cp‖u0‖pq

∫ t

t/2

(t− s)−(n/2)(p/m−1/r)s−p(n/2)(1/q−1/m) ds

≤ C‖u0‖qt−αr

×(1 + t1−n(p−1)/(2q)

∫ 1

1/2

(1− s)−(n/2)(p/m−1/r)s−p(n/2)(1/q−1/m) ds)

≤ C‖u0‖qt−αr

provided p/m− 1/r < 2/n. Since p/m− 1/m < 2/n due to m ≥ q, the conclusionfollows.

Step 6. Positivity. The positivity statement follows from the nonnegativity ofthe semigroup e−tA and the construction of the solution as a limit of nonnegativeiterations uk+1 = Φu0(uk), u1(t) ≡ 0.

In view of the proof of Theorem 15.3, we prepare the following lemma from[535] (see also [529]). It implies in particular a (weighted) a priori estimate for anylocal nonnegative (integral) solution of (15.1) (see Corollary 15.8), which will beused in the proof of Theorem 18.3. Given a measurable function Φ : Ω → [0,∞],we set

(e−tAΦ)(x) :=∫

Ω

G(x, y, t)Φ(y) dy,

where G = GΩ is the Dirichlet heat kernel in Ω (see Appendix B).

Lemma 15.6. Let u0 : Ω → [0,∞] and u : Ω× [0, T ]→ [0,∞] be measurable andsatisfy

u(t) ≥ e−tAu0 +∫ t

0

e−(t−s)Aup(s) ds a.e. in QT . (15.21)

Assume that u(x, t) < ∞ for a.a. (x, t) ∈ QT . Then there holds

t1/(p−1)‖e−tAu0‖∞ ≤ kp := (p− 1)−1/(p−1) for all t ∈ (0, T ]. (15.22)

Proof. In this proof, operations such as interchange of integrals and moving ofe−tA inside integrals are justified by Fubini’s theorem for nonnegative measurablefunctions. First notice that

e−tAΦ = e−(t−s)Ae−sAΦ for all 0 < s < t

and any measurable Φ : Ω → [0,∞].(15.23)

Also, we deduce from Jensen’s inequality and∫Ω G(x, y, t) dy ≤ 1 that

e−tAΦp ≥ (e−tAΦ)p for all measurable Φ : Ω → [0,∞]. (15.24)

Now, by redefining u on a null set, we may assume that (15.21) actually holdseverywhere in Ω× (0, T ). By assumption, for a.a. τ ∈ (0, T ), we have u(·, τ) < ∞a.e. in Ω. Fix such τ and let Ωτ := x ∈ Ω : u(x, τ) < ∞. For t ∈ [0, τ ], it followsfrom (15.21), (15.23) and (15.24) that

e−(τ−t)Au(t) ≥ e−τAu0 +∫ t

0

e−(τ−s)Aup(s) ds

≥ e−τAu0 +∫ t

0

(e−(τ−s)Au(s)

)pds =: h(·, t).

(15.25)

By the second inequality in (15.25), we see that

h(·, t) ≤ e−τAu0 +∫ τ

0

e−(τ−s)Aup(s) ds ≤ u(·, τ); (15.26)

and so h(x, t) < ∞ for all (x, t) ∈ Ωτ × [0, τ ]. Fix x ∈ Ωτ . Then the functionφ(t) := h(x, t) is absolutely continuous on [0, τ ] and (15.25) yields

φ′(t) =(e−(τ−t)Au(t)

)p(x) ≥ φp(t) for a.a. t ∈ [0, τ ]. (15.27)

Also φ(t) ≥(e−τAu0

)(x) > 0, and so (15.27) can be rewritten as [φ1−p]′ ≤ −(p−1).

Integrating this inequality over [0, τ ], we get[(e−τAu0

)(x)]1−p = φ1−p(0) ≥ φ1−p(τ) + (p− 1)τ ≥ (p− 1)τ. (15.28)

It follows that τ1/(p−1)‖e−τAu0‖∞ ≤ k. This guarantees in particular that e−tAu0

∈ L∞(Ω) for a.a. t ∈ (0, T ), Since t → ‖e−tAv‖∞ is continuous for v ∈ L∞(Ω) andt > 0, we deduce from (15.23) that the function

t → t1/(p−1)‖e−tAu0‖∞

is continuous in (0, T ), hence (15.22).


Remark 15.7. If 0 ≤ u0 ∈ L∞(Ω), and u is a (sufficiently regular) supersolutionof (14.1) on [0, T ], then estimate (15.22) can be alternatively obtained as follows(cf. [363]). Let

u(x, t) :=[(

e−tAu0

)1−p(x)− (p− 1)t]−1/(p−1)

+,

which is finite in QT1 , where T1 := inft ∈ [0, T ] : t1/(p−1)‖e−tAu0‖∞ ≥ kp ∈(0, T ]. A direct computation reveals that ut − ∆u ≤ up in QT1 . In view of thecomparison principle, since u = 0 on ST1 and u(·, 0) = u0, we obtain the lowerestimate

u ≥ u in QT1 . (15.29)

In particular, we have T1 = T , hence (15.22).

On the other hand, let us observe that estimate (15.29) also follows from (15.26)and (15.28).

Corollary 15.8. Assume that (15.21) is true with the inequality sign replaced bythe equality sign. Then

‖t1/(p−1)e−tAu(τ)‖∞ ≤ kp for all t ∈ (0, T − τ ] and a.a. τ ∈ (0, T ).

Proof. Set v(t) := u(t + τ). Then (15.23) and Fubini’s theorem guarantee, fora.a. τ ∈ (0, T ) and a.a. t ∈ (τ, T ),

v(t) = e−(t+τ)Au0 +∫ t+τ

0

e−(t+τ−s)Aup(s) ds

= e−tAe−τAu0 +∫ τ

0

e−tAe−(τ−s)Aup(s) ds +∫ t+τ

τ

e−(t+τ−s)Aup(s) ds

= e−tA(e−τAu0 +

∫ τ

0

e−(τ−s)Aup(s) ds)

+∫ t

0

e−(t−s)Avp(s) ds

= e−tAu(τ) +∫ t

0

e−(t−s)Avp(s) ds.

Hence, we may use Lemma 15.6 with u0 replaced by u(τ) and T replaced by T − τfor a.a. τ ∈ (0, T ).

Proof of Theorem 15.3. (i) Fix α ∈ (0, n/q), assume (without loss of generality)that B(0, 2ρ) ⊂ Ω, ρ > 0, and define

u0(y) = |y|−αχB(0,ρ)(y).

Clearly, we have 0 ≤ u0 ∈ Lq(Ω). Using the heat kernel estimate in Proposi-tion 49.10, we obtain, for t > 0 small,

(e−tAu0

)(0) =

∫|y|<ρ

G(0, y, t)|y|−α dy

≥ c1t−n/2

∫√t/2<|y|<√

t|y|−α dy ≥ ct−α/2.

(15.30)

Taking α close enough to n/q, we have α/2 > 1/(p− 1). Combining Lemma 15.6and (15.30), it follows that (15.1) cannot have any integral solution on [0, T ] (cf. Re-mark 15.4(iv)) for any T > 0.

(ii) The assertion is a consequence of Proposition 28.1 below.

For certain applications (see Section 26), it is useful to study well-posedness andregularization properties in different types of Lebesgue spaces. We first considerbounded domains and the spaces Lq

δ(Ω), the Lebesgue spaces weighted by thefunction distance to the boundary. Based on the linear theory in these spaces(see Theorem 49.7 in Appendix C), we obtain the following results [200], in acompletely similar manner as in Theorems 15.2 and 15.3(i). They show that thecritical exponent for local well-posedness is now q = (n + 1)(p− 1)/2.

Theorem 15.9. Assume Ω bounded and p > 1. Let u0 ∈ Lqδ(Ω), 1 ≤ q < ∞,

q > (n + 1)(p − 1)/2. Then there exists T = T (‖u0‖q,δ) > 0 such that problem(15.1) possesses a unique classical Lq

δ-solution in [0, T ) and the following smoothingestimate is true:

‖u(t)‖r,δ ≤ C‖u0‖q,δt−βr , βr :=

n + 12

(1q− 1

r

), (15.31)

for all t ∈ (0, T ] and r ∈ [q,∞], with C = C(n, p, q, Ω) > 0. In addition, u ≥ 0provided u0 ≥ 0.

Theorem 15.10. Assume Ω bounded,

p > 1 +2

n + 1and 1 ≤ q <

(n + 1)(p− 1)2

.

Then there exists u0 ∈ Lqδ(Ω), such that (15.1) does not admit any nonnegative

Lqδ-solution in [0, T ) for any T > 0.

In the case Ω = Rn, let us finally consider the uniformly local Lebesgue spaces

Lqul(R

n). Using the linear smoothing effect in these spaces (see Proposition 49.15in Appendix C or [37]), we obtain the following smoothing estimate (cf. [253]) bysimilar arguments as in the proof of Theorem 15.2. Here e−tA denotes the heatsemigroup in R

n.

16. Maximal existence time. Uniform bounds from Lq-estimates 87

Theorem 15.11. Let p > 1, q > qc and 1 ≤ q < ∞. Let u0 ∈ L∞(Rn), T > 0and assume that u ∈ L∞((0, T ), L∞(Rn)) is a solution of

u(t) = e−tAu0 +∫ t

0

e−(t−s)A|u(s)|p−1u(s) ds, 0 < t < T.

Then there exist C = C(n, p, q) > 0, T0 = T0(‖u0‖q,ul) > 0 such that

‖u(t)‖r,ul ≤ C‖u0‖q,ult−αr , αr :=

n

2

(1q− 1

r

),

for all t ∈ (0, min(T, T0)] and r ∈ [q,∞].

Remark 15.12. A local well-posedness result similar to Theorem 15.2 can alsobe proved in Lq

ul(Rn) (see [253], and cf. also [28]).

16. Maximal existence time. Uniform boundsfrom Lq-estimates

In this section we are interested in sufficient conditions guaranteeing global exis-tence. More precisely, we want to show that any solution satisfying suitable boundsin the Lebesgue space Lq(Ω) is global.

Let us start with a simple proposition which defines the maximal solution andexistence time. We formulate the statement only for the model problem (14.1) butit will be clear from the proof that the same statement is true for a much moregeneral class of equations and systems.

Proposition 16.1. Let X be a Banach space of functions defined in Ω. Assumethat problem (14.1) possesses for each u0 ∈ X a unique (classical X-) solution u onthe interval [0, T ], where T = T (u0). Then there exists Tmax = Tmax(u0) ∈ (T,∞]with the following properties.(i) The solution u can be continued (in a unique way) to a classical X-solution onthe interval [0, Tmax).(ii) If Tmax < ∞, then u cannot be continued to a classical X-solution on [0, τ)for any τ > Tmax. We call u the maximal (classical X-) solution starting from u0

and Tmax its maximal existence time.(iii) Assume further that T = T (‖u0‖X). Then

either Tmax = ∞ or limt→Tmax ‖u(t)‖X = ∞. (16.1)

Proof. Let u0 ∈ X be fixed. If u1 and u2 are solutions of (14.1) on [0, T1) and[0, T2), respectively, then u1 = u2 on [0, min(T1, T2)) due to the uniqueness. Let

uα : [0, Tα) → X be the set of all solutions of (14.1) and T := sup Tα. Defineu : [0, T ) → X by u(t) := uα(t), where α is any index such that Tα > t. Then u isobviously a solution of (14.1) on [0, T ), and properties (i)(ii) are verified.

Under the assumption in property (iii), suppose that

T < ∞ and lim inft→T

‖u(t)‖X < ∞.

Choose C > 0 and tk → T such that ‖u(tk)‖X < C for all k = 1, 2, . . . . Due toour assumptions there exists T > 0 independent of k such that the problem (14.1)with initial data u(tk) possesses a unique solution uk : [0, T ] → X , k = 1, 2, . . . .By uniqueness, uk(t) = u(t + tk) for t small. Fix k such that tk ∈ (T − T, T ) andset

u(t) :=

u(t), t ∈ [0, tk],uk(t− tk), t ∈ [tk, tk + T ].

Then u is a solution of (14.1) on [0, tk + T ] and tk + T > T which contradicts thedefinition of T .

Remarks 16.2. (i) Maximal Lq-solution. Consider problem (15.1) and setX = Lq(Ω), where 1 ≤ q ≤ ∞ satisfies q > qc = n(p − 1)/2 or q = qc > 1. Ifu0 ∈ X , then Theorem 15.2 and Proposition 16.1 (or Remark 51.11 if q = ∞)guarantee the existence of a maximal (classical Lq-) solution u, up to a maximalexistence time Tmax(u0). Moreover, property (16.1) is true if q > qc. Similarly asin Example 51.9, u in fact satisfies

u ∈ BC2,1(Ω× [t1, t2]), 0 < t1 < t2 < Tmax(u0). (16.2)

If u0 ≥ 0, then u ≥ 0. If u0 is radial (resp. nonnegative and radial nonincreasing),then u enjoys the same property, as a consequence of Proposition 52.17.

(ii) Independence of the maximal solution with respect to q. Let q, u0

and u be as in remark (i). We show that, if u0 belongs to several Lq-spaces, thenu and Tmax(u0) do not depend on q.

Thus assume that u0 ∈ Lq1 ∩ Lq2(Ω) for some q1, q2 as above, and denoteby ui, i = 1, 2, the corresponding maximal, classical Lqi-solution, of existencetime T i. We know that u1 = u2 for t > 0 small (cf. Remark 15.5). Using ui ∈C([0, T i), Lqi(Ω)), we deduce easily that u1 = u2 on [0, min(T 1, T 2)). Assume forcontradiction that T 1 < T 2 (hence T 1 < ∞). Since, by the definition of a maximalclassical Lq-solution, u2 ∈ L∞

loc((0, T 2), L∞(Ω)), it follows that ||u1|p−1u1| ≤ C|u1|on (T 1/2, T 1), which readily implies sup[T 1/2,T 1] ‖u1(t)‖q1 < ∞. If q1 > qc, thenthis contradicts (16.1). If q = qc, then the variation-of-constants formula impliesu1 ∈ C([0, T 1], Lq1(Ω)). The local existence theorem can then be used to extendu1 after T 1 and we again reach a contradiction.

(iii) Lower bounds on supercritical Lq-norms. By using the local theoryof problem (14.1), developed in Section 15, it is actually possible to obtain lower

16. Maximal existence time. Uniform bounds from Lq-estimates 89

estimates of the supercritical Lq-norms, in case Tmax(u0) < ∞. Namely, let q ≥ 1satisfy qc < q < ∞ and assume u0 ∈ Lq(Ω). Then the proof of Theorem 15.2(see in particular formula (15.14)) shows that (Tmax(u0))1−n(p−1)/2q‖u0‖p−1

q ≥C(n, p, q) > 0. After a time shift, this yields

‖u(t)‖q ≥ C(n, p, q)(Tmax(u0)− t)n/(2q)−1/(p−1), 0 ≤ t < T. (16.3)

(iv) Critical Lq-space. If u0 ∈ Lq(Ω) with q = qc > 1, then it is not knownin general whether Tmax(u0) < ∞ implies limt→Tmax(u0) ‖u(t)‖q = ∞. A positiveresult in that direction (cf. [91], [535]) is given by the following proposition. Theproof, which relies on simple energy arguments, is postponed to the next section.For further positive results, see [530], [535], [246], [357].

Proposition 16.3. Consider problem (15.1) with p = 1 + 4/n (so that qc =n(p− 1)/2 = 2). Let u0 ∈ L2(Ω) and assume T := Tmax(u0) <∞. Then

‖u(t)‖2 ≥ C(n, p)| log(T − t)|1/2, t→ T. (16.4)

We say that (14.1) possesses a global solution if Tmax = ∞. Proposition 16.1provides a simple criterion for global existence: If ‖u(t)‖X remains bounded, thenTmax = ∞. Since the assumptions of Proposition 16.1 are satisfied with X =L∞(Ω) if f ∈ C1 (see Remark 51.11) we see that the boundedness of the solutionin L∞(Ω) is sufficient for its global existence. Note that the same statement is truefor a much more general class of equations and systems.

Unfortunately, it is not easy to obtain the L∞-estimate for solutions of (14.1).As we shall see, standard methods usually yield only an Lq-estimate for someq < ∞. Therefore, it is important to find q as small as possible and such that theLq-estimate guarantees the L∞-estimate, hence global existence. We will call thisproperty of Lq the continuation property.

Theorem 15.2, Proposition 16.1 and Remarks 16.2 guarantee the global existenceof a solution of the model problem (15.1) provided the solution is bounded in Lq(Ω)for some q > qc.

As we shall see in Corollary 24.2, this condition is optimal (up to the equalitysign): If 1 < q < qc, then there exists a radial positive solution of (15.1) in aball such that Tmax <∞ but the solution stays bounded in Lq(Ω). Therefore, theexponent qc for problem (15.1) is “critical” both for well-posedness and the globalexistence. This is due to the simple structure of the nonlinearity. We will see inChapter III that for more complicated problems, the critical exponents for the well-posedness and the continuation property may differ. Therefore it is important tofind methods guaranteeing the global existence of a solution under the assumptionof its boundedness in Lq and not using any well-posedness result.

In this section we present a method due to [11], [12] (cf. also [459]) which isbased on Moser-type iterations and can be efficiently used for a very general class

of problems (including degenerate problems, problems on nonsmooth domains etc).In order to make it as clear as possible, we again restrict ourselves to the modelproblem (15.1).

Another method for obtaining L∞-bounds from Lq-bounds is presented in Ap-pendix E (see Proposition 51.34). That method is based on the variation-of-constants formula and interpolation-extrapolation spaces and is due to [14].

Hence, our aim is to prove the following theorem (which is a consequence ofTheorem 15.2 and Proposition 16.1), without using the well-posedness results.

Theorem 16.4. Let p > 1 and let u be a classical solution of (15.1) defined on[0, T ). Assume q > 1 and Uq := supt<T ‖u(t)‖q < ∞. If p < 1 + 2q/n, thenU∞ < ∞.

For simplicity we will assume that Ω is bounded and n ≥ 3.

Lemma 16.5. Let u be a classical solution of (15.1) on [0, T ), r ≥ q ≥ 1, p <

1 + 2q/n and Ur := max1, ‖u0‖∞, supt<T ‖u(t)‖r < ∞. Set

σ(r) :=n + 22n

(2r

n+ 1− p

)−1

and ρ(r) := 1 + (p− 1)σ(r).

Then there exists a constant C1 = C1(p, q, n, Ω) > 0 such that

U2r ≤ C1/r1 rσ(r)Uρ(r)

r .

Proof. Multiplying the equation in (15.1) by |u|2r−2u we obtain

d

dt

12r

∫Ω

|u|2r dx +2r − 1

r2

∫Ω

∣∣∇|u|r∣∣2 dx =∫

Ω

|u|p+2r−1 dx.

Denote w := |u|r, α = α(r) := (p + 2r− 1)/(2r) and let β be defined by 1/(2α) =β + (1− β)/2∗. Then the assumption 1 < p < 1 + 2r/n guarantees β ∈ (0, 1) andα(1 − β) < 1. The above identity, interpolation, the Sobolev embedding theoremand Young’s inequality imply

d

dt

12r‖w‖22 +

2r − 1r2

‖∇w‖22 = ‖w‖2α2α ≤

(‖w‖β

1‖w‖1−β2∗)2α

≤ C(‖w‖β

1‖∇w‖1−β2

)2α=( 1

2r‖∇w‖22

)α(1−β)(Cr1−β‖w‖2β

1

)α

≤ 12r‖∇w‖22 + Crα(1−β)/δ‖w‖2αβ/δ

1 =12r‖∇w‖22 + Cr2rσ(r)−1‖w‖2ρ(r)

1 ,

where δ := 1− α(1 − β). Consequently, there exist C, c > 0 such that

e−ct d

dt

(ect‖w‖22

)=

d

dt‖w‖22 + c‖w‖22 ≤ Cr2rσ(r)‖w‖2ρ(r)

1 ≤ Cr2rσ(r)U2rρ(r)r .

17. Blow-up 91

Since ‖w(0)‖22 ≤ C‖u0‖2r∞ ≤ CU2r

r and ‖w‖22 = ‖u‖2r2r, integration of the above

estimate implies the assertion.

Proof of Theorem 16.4. We shall use the notation from Lemma 16.5. Noticethat γ := qσ(q) ≥ rσ(r) for any r ≥ q. Using repeatedly Lemma 16.5 with r := q,r := 2q, r := 4q etc, one can easily verify that, given ν ∈ 0, 1, 2, . . .,

U2ν+1q ≤ (C1qγ)k12k2Uk3

q ,

where

k1 = k1(ν) =1

2νq+

ρ(2νq)2ν−1q

+ · · ·+ ρ(2νq) · · · · · ρ(2q)q

,

k2 = k2(ν) =γ

q

( ν

2ν+

ν − 12ν−1

ρ(2νq) + · · ·+ 12ρ(2νq) · · · · · ρ(22q)

),

k3 = k3(ν) = ρ(2νq) · · · · · ρ(q).

Since σ(2r) ≤ σ(r)/2 we obtain ρ(2iq) ≤ 1 + (p − 1)σ(q)2−i. Now using theinequality log(1 + x) ≤ x for x ≥ 0 we get

log k3 ≤ν∑

i=0

(p− 1)σ(q)2i

≤ 2(p− 1)σ(q) =: C3 < ∞.

Finally,

k1 ≤k3

q

ν∑i=0

12i≤ 2

qeC3 < ∞ and k2 ≤

γ

qk3

ν∑i=1

i

2i≤ γ

qeC3

∞∑i=1

i

2i< ∞,

which concludes the proof.

17. Blow-up

In this section we mainly consider the model problem

ut −∆u = λu + |u|p−1u, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (17.1)

with p > 1 and λ ∈ R, and we derive some criteria for u0 which guarantee blow-upof the solution of (17.1) in finite time. More general nonlinearities f(u) will bebriefly considered.

We will always assume that u0 belongs to a function space X where (17.1) iswell-posed and we denote by Tmax(u0) the maximal existence time of the solutionof (17.1). We start with a simple criterion. In the bounded domain case, it is basedon the eigenfunction method due to [296].

Theorem 17.1. Consider problem (17.1) with p > 1 and λ ∈ R.(i) Assume Ω bounded. Let u0 ∈ L∞(Ω) satisfy

u0 ≥ 0,

∫Ω

u0ϕ1 dx > c :=(max(0, λ1 − λ)

)1/(p−1). (17.2)

Then Tmax(u0) < ∞.(ii) Assume Ω = R

n. Then assertion (i) remains valid if we replace λ1 by 2n andϕ1 by the function ϕ(x) = π−n/2e−|x|2.

Proof. (i) Recall that u ≥ 0 and denote y = y(t) :=∫Ω u(t)ϕ1 dx. Multiplying

the equation in (17.1) with ϕ1, integrating by parts, and using ∆ϕ1 = −λ1ϕ1 andJensen’s inequality yields

y′ =∫

Ω

utϕ1 dx =∫

Ω

u∆ϕ1 dx + λ

∫Ω

uϕ1 dx +∫

Ω

upϕ1 dx ≥ yp − cp−1y. (17.3)

Since y(0) > c, we infer from (17.3) that

y′ ≥ εyp, 0 < t < Tmax(u0),

with ε = 1− (c/y(0))p−1 > 0. This differential inequality guarantees that u cannotexist globally.

(ii) The proof is the same except that we now use ∆ϕ ≥ −2nϕ. The calculationin (17.3) can be easily justified by integrating by parts over BR and letting R →∞,using property (16.2) and the exponential decay of ϕ and ∇ϕ.

Remarks 17.2. (i) Estimation of the blow-up time. The proof of Theo-rem 17.1 shows that if, for instance, λ = 0, Ω is bounded and

∫Ω

u0ϕ1 dx ≥(2λ1)1/(p−1), then

Tmax(u0) ≤2

p− 1

(∫Ω

u0ϕ1 dx)1−p

.

We refer to [324], [262] for more precise results concerning upper estimates of theblow-up time.

(ii) Neumann boundary conditions. If we replace the homogeneous Dirichletboundary conditions in problem (17.1) with the homogeneous Neumann boundaryconditions ∂νu = 0, then all positive solutions blow up in finite time when λ ≥0. Indeed, by integrating the equation over Ω, we see that the function y(t) :=∫Ω

u(t) dx satisfies y′(t) ≥∫Ω

up dx ≥ |Ω|1−pyp with y(0) > 0. Alternatively, fort0 > 0 small, the strong maximum principle guarantees that u(x, t0) ≥ ε > 0 in Ω,and it suffices to use the solution of the ODE z′ = zp, z(t0) = ε, as subsolution.

The previous result can be easily extended to problem (14.1) under suitableconvexity and superlinearity conditions.

17. Blow-up 93

Theorem 17.3. Consider problem (14.1) where f : R → R is a convex C1-function and Ω is bounded. Assume that, for some a > 0, we have f(s) > 0 for alls ≥ a and ∫ ∞ ds

f(s)< ∞. (17.4)

Then Theorem 17.1(i) remains true provided the constant c in (17.2) is replacedby C = C(Ω, f) > 0 large enough.

Proof. Denote again y = y(t) :=∫Ω u(t)ϕ1 dx. Arguing as in the previous proof,

we obtain

y′ =∫

Ω

utϕ1 dx =∫

Ω

u∆ϕ1 dx +∫

Ω

f(u)ϕ1 dx ≥ −λ1y + f(y). (17.5)

Since f is convex, the function g(s) := f(s)−f(a)s−a is nondecreasing for s > a and

g(s) → ∞ as s → ∞, due to (17.4). Therefore, there exists C ≥ a such thatf(s) ≥ 2λ1s for all s ≥ C. If y(0) ≥ C, it follows from (17.5) that, as long as uexists, y(t) ≥ C and

y′ ≥ f(y)− λ1y ≥12f(y),

hence

λ1t/2 ≤∫ t

0

y′(τ)f(y(τ))

dτ =∫ y(t)

y(0)

ds

f(s)≤∫ ∞

y(0)

ds

f(s)< ∞.

Therefore u cannot exist globally.

Remark 17.4. It is well known that condition (17.4) is necessary and sufficientfor the existence of blow-up solutions of the ODE u′ = f(u), t ≥ 0. The convexitycondition in Theorem 17.3 can be replaced by the assumption that f ≥ f fors large, where f satisfies the assumptions of the theorem. As a typical “weaklysuperlinear” f satisfying (17.4), one may take a function f such that f(s) =(1 + s) logp(1 + s) for s ≥ 0, with p > 1.

The next criterion is based on the fact that the energy functional

E(u) =12

∫Ω

(|∇u|2 − λu2

)dx − 1

p + 1

∫Ω

|u|p+1 dx (17.6)

is nonincreasing along any solution of (17.1). More precisely we have:

Lemma 17.5. Consider problem (17.1) with p > 1, λ ∈ R, u0 ∈ L∞ ∩ H10 (Ω),

and let T = Tmax(u0). Then E(u(·)) ∈ C([0, T )) ∩ C1((0, T )) and

d

dtE(u(t)

)= −

∫Ω

u2t (t) dx. (17.7)

Proof. Example 51.28 guarantees u ∈ C([0, T ), H1

0 ∩ Lp+1(Ω)), hence E(u(·)) ∈

C([0, T )), and

u ∈ C((0, T ), H2(Ω)

)∩C1

((0, T ), L2 ∩ Lp+1(Ω)

). (17.8)

Denote E1(t) =∫Ω |∇u(t)|2 dx. For t, s ∈ (0, T ), s = t, using integration by parts,

we obtain

E1(t)− E1(s)t− s

=1

t− s

∫Ω

∇(u(t)− u(s)) · ∇(u(t) + u(s)) dx

= −∫

Ω

(u(t)− u(s)t− s

)∆(u(t) + u(s)) dx → −2

∫Ω

ut(t)∆u(t) dx

as s → t, due to (17.8). Consequently, E(u(·)) ∈ C1((0, T )) and

d

dtE(u(t)

)=∫

Ω

(−∆u− λu− |u|p−1u)ut dx = −∫

Ω

u2t (t) dx.

The following result is due to [327].The simpler proof in the case Ω bounded, λ = 0, is from [515], [51].

Theorem 17.6. Consider problem (17.1) with p > 1, λ ∈ R and u0 ∈ L∞∩H10 (Ω).

Assume either Ω bounded or λ ≤ 0. If E(u0) < 0, then Tmax(u0) <∞.

Proof. (i) First assume that Ω is bounded.Set ψ(t) := ‖u(t)‖22. Multiplying the equation in (17.1) by u and using Holder’s

inequality we obtain

12ψ′(t) =

∫Ω

uut(t) dx = −∫

Ω

|∇u(t)|2 dx + λ

∫Ω

u2 dx +∫

Ω

|u(t)|p+1 dx

= −2E(u(t)

)+

p− 1p + 1

∫Ω

|u(t)|p+1 dx ≥ −2E(u0) + cψ(t)(p+1)/2,

(17.9)

where c := (p − 1)/[(p + 1)|Ω|(p−1)/2]. This inequality implies Tmax(u0) < ∞provided E(u0) < 0 (or ψ(0)(p+1)/2 > 2E(u0)/c).

(ii) Next consider the case Ω unbounded, λ ≤ 0. (The following argument worksalso if λ ≤ λ1 and Ω is bounded). We will use the concavity method due to[327].

Assume Tmax(u0) = ∞ and denote M(t) := 12

∫ t

0‖u(s)‖22 ds. Then we have

M ′(t) = 12‖u(t)‖22 and

M ′′(t) =∫

Ω

uut(t) dx = −∫

Ω

|∇u(t)|2 dx + λ

∫Ω

u2(t) dx +∫

Ω

|u(t)|p+1 dx

= −(p + 1)E(u(t)

)+

p− 12

∫Ω

(|∇u(t)|2 − λu2(t)

)dx

≥ −(p + 1)E(u0) > 0,

17. Blow-up 95

which implies M ′(t) →∞ and M(t)→∞ as t→∞. Moreover, this estimate and

∫ t

0

‖ut(s)‖22 ds = E(u0)− E(u(t)

)< −E

(u(t)

)(17.10)

(cf. (17.7)) imply

M ′′(t) ≥ −(p + 1)E(u(t)

)≥ (p + 1)

∫ t

0

‖ut(s)‖22 ds,

hence

M(t)M ′′(t) ≥ p + 12

(∫ t

0

‖ut(s)‖22 ds)(∫ t

0

‖u(s)‖22 ds)

≥ p + 12

(∫ t

0

∫Ω

u(x, s)ut(x, s) dx ds)2

=p + 1

2(M ′(t)−M ′(0)

)2.

Since M ′(t) → ∞ as t → ∞, the last estimate implies existence of α, t0 > 0 suchthat

M(t)M ′′(t) ≥ (1 + α)(M ′(t)

)2, t ≥ t0.

This inequality guarantees that the nonincreasing function t →M−α(t) is concaveon [t0,∞) which contradicts the fact M−α(t) → 0 as t→∞.

Proof of Proposition 16.3. By Example 51.27 in Appendix E, after a time-shift, we may assume u0 ∈ H1

0 (Ω). Similarly as in (17.9) (but without assuming Ωbounded), we have

12

d

dt

∫Ω

u2(t) dx ≥ −2E(u0) +p− 1p + 1

∫Ω

|u(t)|p+1 dx.

Integrating and using (16.3) with q = p + 1 > qc, it follows that

∫Ω

u2(t) dx ≥ −2E(u0)t +p− 1p + 1

∫ t

0

∫Ω

|u(s)|p+1 dx ds

≥ −C + C(n, p)∫ t

0

(T − s)(n/2)−(p+1)/(p−1) ds.

Since (n/2)− (p + 1)/(p− 1) = −1, (16.4) follows.

Remarks 17.7. (i) The proof of Theorem 17.6 does not imply blow-up of theL2-norm of u. Indeed, as was observed in [51], the solution u might cease toexist before the time obtained by integrating the differential inequality in (17.9).Examples where the L2-norm of u remains bounded will be given in Corollary 24.2.A similar remark holds concerning the quantity y(t) in the proof of Theorem 17.1.

(ii) The first part of the proof of Theorem 17.6 shows that

‖u(t)‖2 ≤(2E(u0)/c

)1/(p+1)

for any global solution u of (17.1) provided Ω is bounded. Now the results of thepreceding section guarantee that ‖u(t)‖∞ ≤ C

(‖u0‖∞, E(u0)

)if p < 1 + 4/n. As

we shall see later in Section 22, this assertion is true for any p < pS .

(iii) If u is a global solution of (17.1) and Ω is bounded or λ ≤ 0, then Theo-rem 17.6 guarantees 0 ≤ E

(u(t)

)≤ E(u0) for all t > 0.

(iv) Inequality (17.9) also shows the following: Given δ > 0 there exists Cδ > 0such that Tmax(u0) < δ whenever E(u0) < −Cδ.

(v) Let ϕ ∈ L∞ ∩H10 (Ω) be a fixed function, ϕ ≡ 0. Then Tmax(αϕ) < ∞ for

α > 0 large enough. This follows from Theorem 17.6 and the fact that

E(αϕ) = α2

∫Ω

|∇ϕ|2 − λϕ2

2dx − αp+1

∫Ω

ϕp+1

p + 1dx.

Note that if we assume 0 ≤ ϕ ∈ L∞(Ω) instead, then the same conclusion followsfrom Theorem 17.1.

Further blow-up conditions involving the energy will be given in Theorem 19.5.We now give a third criterion (cf. [342]), which guarantees blow-up if one startsabove a positive equilibrium.

Theorem 17.8. Assume Ω bounded, p > 1 and λ ∈ R. Assume that problem(17.1) has a (classical) equilibrium v, with v > 0 in Ω. If u0 ∈ L∞(Ω) satisfiesu0 ≥ v, u0 ≡ v, then Tmax(u0) < ∞.

For the proof, we prepare the following separation lemma, which will be usedagain later.

Lemma 17.9. Assume Ω bounded and consider problem (14.1) where f : R → R

is a convex C1-function with f(0) = 0. Let u0, u0 ∈ L∞(Ω) be such that u0 ≥ u0,u0 ≡ u0. Let u, u be the corresponding solutions of (14.1), and fix τ ∈ (0, Tmax(u0)).Then Tmax(u0) ≥ Tmax(u0) and there exists α > 1 such that

u ≥ αu, τ ≤ t < Tmax(u0). (17.11)

17. Blow-up 97

Proof. Since u ≤ u by the comparison principle and f(s) ≥ f ′(0)s, s ∈ R, by theconvexity of f and f(0) = 0, we have Tmax(u0) ≥ Tmax(u0). By the strong and theHopf maximum principles, we have

u(x, τ) > u(x, τ) in Ω and∂u

∂ν(x, τ) <

∂u

∂ν(x, τ) on ∂Ω.

Therefore, there exists α > 1 such that u(x, τ) ≥ αu(x, τ) in Ω. Since f(αu) ≥αf(u), due to f convex and f(0) = 0, we infer that

(αu)t −∆(αu)− f(αu) ≤ α(ut −∆u− f(u)) = 0,

and the lemma follows from the comparison principle.

Proof of Theorem 17.8. By Lemma 17.9, applied with u0 = v, there exist α > 1and τ ∈ (0, Tmax(u0)), such that

u ≥ αv, t ∈ [τ, Tmax(u0)). (17.12)

Denote z = z(t) :=∫Ω u(t)v dx. Multiplying the equation in (15.1) with v, in-

tegrating by parts, and using (17.12) and Holder’s (or Jensen’s) inequality, weobtain

z′ =∫

Ω

utv dx =∫

Ω

u∆v dx +∫

Ω

(up + λu)v dx

=∫

Ω

(upv − vpu)dx =∫

Ω

(1− (v/u)p−1

)upv dx

≥ (1− α1−p)∫

Ω

upv dx ≥ (1− α1−p)(∫

Ω

v dx)1−p

zp,

for t ∈ [τ, Tmax(u0)). It follows that u cannot exist globally.

By using an alternative linearization argument based on an idea from [311], onecan extend Theorem 17.10 to more general convex nonlinearities.

Theorem 17.10. Consider problem (14.1) with f and Ω as in Theorem 17.3.Assume in addition that f(0) = 0, f ′ is nonconstant near 0, and that problem(14.1) has a (classical) equilibrium v, with v > 0 in Ω. If u0 ∈ L∞(Ω) satisfiesu0 ≥ v, u0 ≡ v, then Tmax(u0) < ∞.

Proof. Let µ and ψ > 0 denote the first eigenvalue and the corresponding eigen-function of the problem

∆ψ + f ′(v)ψ = µψ in Ω, ψ = 0 on ∂Ω,∫Ω

ψ dx = 1. Multiplying the above equation by v, the equation ∆v + f(v) = 0 byψ, integrating and subtracting the resulting identities, we obtain

µ

∫Ω

vψ dx =∫

Ω

(vf ′(v) − f(v))ψ dx.

Due to v, ψ > 0, f(0) = 0, f convex and f ′ nonconstant near 0, the last integralis positive, hence µ > 0 (the solution v is linearly unstable).

Since u ≥ v by the comparison principle, we have y(t) :=∫Ω(u(t)− v)ψ dx ≥ 0.

In addition,

y′(t) =∫

Ω

(∆u + f(u))ψ dx =∫

Ω

((u− v)∆ψ + (f(u)− f(v))ψ

)dx

=∫

Ω

µ(u− v)ψ +(f(u)− f(v)− f ′(v)(u − v)

)ψ dx.

Since f(u)− f(v)− f ′(v)(u− v) ≥ 0 by convexity, we have y′(t) ≥ µy(t). Assumefor contradiction that Tmax(u0) = ∞. Then limt→∞ y(t) = ∞. Since ψ ≤ cϕ1

due to (1.4), it follows that limt→∞∫Ω

u(t)ϕ1 dx = ∞. But this contradicts Theo-rem 17.3.

Remark 17.11. The proofs of Theorems 17.8 and 17.10 were based on the con-vexity of the nonlinearity. However, if v is a maximal, unstable equilibrium, thenblow-up of solutions starting above v can be shown for general superlinear f withsubcritical growth.

Assume firstf(cu) ≥ cf(u) for c > 1 (17.13)

and let u0 ≥ v, u0 ≡ v. Fix τ > 0. Due to the maximum principle, there existsε > 0 such that u(τ) ≥ (1 + ε)v =: u0 (cf. the proof of Lemma 17.9). Let u denotethe solution with initial data u0. Since u(t + τ) ≥ u(t) by the maximum principle,it suffices to prove Tmax(u0) < ∞. Assume on the contrary that u exists globally.Since ∆u0 + f(u0) ≥ 0, we have ut ≥ 0. Lemma 53.10 and the maximality of vguarantee that u cannot stay bounded, hence ‖u(t)‖∞ →∞ as t → ∞. Since thegrowth of f is subcritical, we have also ‖u(t)‖1,2 → ∞ as t → ∞. Now a simplemodification of the concavity method (cf. the proof of Theorem 17.6(ii) and see[186] for details) yields a contradiction.

If f is a general function (not necessarily satisfying (17.13)), then [354] guar-antees the existence of a time increasing solution w defined for t ∈ (−∞, 0] andsatisfying w(t) → v in C1(Ω) as t → −∞. Fix τ > 0. Since u(τ) ≥ w(t) for suit-able t ≤ 0 we can proceed as above. This approach can be used for more generalproblems provided one can show boundedness of global increasing solutions (see[185], for example).

Our last criterion [55], [324] concerns the Cauchy problem and asserts thatfinite-time blow-up occurs whenever the nonnegative initial data has a sufficientlyslow decay at infinity.

Theorem 17.12. Let p > 1 and consider problem (15.1) with Ω = Rn. Let

−µ < 0 be the first eigenvalue of the Dirichlet Laplacian in the unit ball of Rn. If

17. Blow-up 99

0 ≤ u0 ∈ L∞(Rn) satisfies

lim inf|x|→∞

|x|2/(p−1)u0(x) > µ1/(p−1), (17.14)

then Tmax(u0) < ∞.

Remarks 17.13. (i) Slow decay in more general domains. A similar resultholds (with a different constant on the RHS of (17.14)) if the inferior limit is takenon a cone Σ, instead of the whole space (see [497]). The proof of [497], is different,based on scaling and comparison arguments. Similar blow-up conditions still holdfor more general domains Σ, typically a paraboloid of the form Σ = x = (x′, xn) ∈R

n : xn > 0, |x′n| < xβ

n for some 0 < β < 1, the power 2/(p− 1) in (17.14) beingthen replaced by the smaller number 2β/(p − 1) (see [386], [461]). Also, similarresults can be proved when Ω itself is replaced by such a domain.

(ii) Sign-changing initial data with slow decay. An extension of Theo-rem 17.12 to sign-changing solutions has been obtained in [386].

Proof of Theorem 17.12. Assume that Tmax(u0) = ∞. For R > 0, denote byλ1,R the first Dirichlet eigenvalue of −∆ in the ball BR. Let ϕ1,R be the corre-sponding eigenfunction satisfying

∫BR

ϕ1,R dx = 1. We know that Tmax(u0) = ∞implies ∫

BR

u0 ϕ1,R dx ≤ λ1/(p−1)1,R . (17.15)

Indeed, this follows from the proof of Theorem 17.1, using the fact that∫BR

ϕ1,R ∆u dx =∫

BR

u ∆ϕ1,R dx−∫

∂BR

u ∂νϕ1,R dσ ≥ −λ1,R

∫BR

u ϕ1,R dx.

Set ψ := ϕ1,1. By standard scaling properties of eigenfunctions and eigenvalues,we have ϕ1,R(x) = R−nψ(R−1x), x ∈ BR, and λ1,R = R−2µ. For each ε ∈ (0, 1),(17.15) implies

µ1

p−1 R− 2p−1 ≥

∫εR<|x|<R

u0(x)ϕ1,R(x) dx

≥(

infεR<|x|<R

u0(x)) ∫

εR<|x|<R

R−nψ(R−1x) dx,

henceµ

1p−1 ≥

(inf

εR<|x|<R|x|

2p−1 u0(x)

) ∫ε<|y|<1

ψ(y) dy.

Setting = lim inf |x|→∞ |x|2/(p−1)u0(x) and letting R →∞, we get

µ1

p−1 ≥

∫ε<|y|<1

ψ(y) dy,

hence µ1

p−1 ≥ upon letting ε→ 0. The result follows.

Remark 17.14. Comparison of domains. Assume that Ω1 ⊂ Ω2 are (possiblyunbounded) smooth domains. Let 0 ≤ u0 ∈ L∞(Ω1) and extend u0 by 0 outsideΩ1. Denote by ui, i ∈ 1, 2, the solution of problem (17.1) in Ω = Ωi withinitial data u0. If u1 is nonglobal, then so is u2 (this follows from the comparisonprinciple applied in Ω1). This simple fact illustrates the heuristic principle that“larger domains are more instable” (cf. [328]). From this and, e.g., Theorem 17.1,one can derive blow-up criteria in general unbounded domains.

Remark 17.15. Quenching. The so-called quenching phenomenon is closelyrelated to blow-up. Instead of f = |u|p−1u or f satisfying condition (17.4), considera “singular” nonlinearity f : [0, a) → [0,∞) for some a ∈ (0,∞), of class C1, andsatisfying

lims→a− f(s) = ∞;

typicallyf(u) = λ(a− u)−p for some λ, p > 0. (17.16)

Assume that 0 ≤ u0 ∈ L∞(Ω) is such that ess supΩ u0 < a. Then problem (14.1)still admits a unique, maximal, classical solution u ≥ 0, and it is easy to show thateither

T := Tmax(u0) =∞, or T <∞ and limt→T

‖u(t)‖∞ = a.

The latter case is called (finite-time) quenching: The solution itself remains bound-ed, but a singularity appears in the RHS. In fact, it can be shown that undersuitable assumptions, quenching implies blow-up of ut, namely limt→T ‖ut(t)‖∞ =∞. Different, but related, is the phenomenon of gradient blow-up; see Sections 40and 41.

If, for instance, f is given by (17.16) with λ large enough, then quenchingoccurs for all u0 as above. The quenching problem, first considered in [303], hasbeen investigated in numerous articles. We refer to, e.g., [329], [120] for surveyson this subject.

18. Fujita-type results

Consider problem (15.1) with Ω = Rn:

ut −∆u = |u|p−1u, x ∈ Rn, t > 0,

u(x, 0) = u0(x), x ∈ Rn.

(18.1)

Assume p > 1 and u0 ≥ 0. The results of the preceding section guarantee that thesolution of (18.1) blows up in finite time if u0 is sufficiently large. In this section

18. Fujita-type results 101

we show that all solutions of (18.1) with u0 ≥ 0, u0 ≡ 0, blow up in finite time ifand only if p ≤ pF , where

pF := 1 +2n

.

The number pF thus plays the role of a critical exponent for the Cauchy problem(18.1). This result was proved in [220] for p = pF and in [271], [307] (see also [35],[530]) for p = pF .

Numerous generalizations and modifications can be found in (the references of)the survey articles [328], [159] and in [372]. Some of them are described in theremarks at the end of this section, in Theorem 32.7, and in Sections 37 and 45.On the other hand, an application to a model arising in population genetics willbe given at the end of this section.

Theorem 18.1. (i) Let 1 pF . Then problem (18.1) has a global, classical solution for somepositive u0 ∈ L∞(Ω).

Remark 18.2. (i) By a distributional solution, we here mean a function u ∈Lp

loc(Q) which satisfies (18.2) inD′(Q). The proof of assertion (i) given below showsthat this remains true for distributional solutions of the inequality ut −∆u ≥ up

in Q. We use a modification of arguments in [372], based on rescalings of a simple,compactly supported test-function, depending on x and t.

A related proof can be found in [56], where the test-functions are obtained bysolving an adjoint problem. The original proof of [220] involved Gaussian test-functions depending on x only (given by the heat kernel with t as a parameter),hence requiring more regularity of the solutions in time.

(ii) In the result of Theorem 18.1(i), the roles played by the behaviors of thenonlinearity as u → 0 and as u → ∞ are different; see Remark 18.8(iii) for de-tails.

Proof of Theorem 18.1(i). Let u ≥ 0 be a distributional solution of (18.2) inQ and fix t0 > 0.

Step 1. We claim that, for each ξ ∈ D(Rn), ψ ∈ C∞([t0,∞)), with ξ, ψ ≥ 0,ψ(t) = 1 near t = t0 and ψ(t) = 0 for t large, there holds∫ ∞

t0

∫Rn

upξψ dx dt ≤ −∫ ∞

t0

∫Rn

u(ξ∂tψ + ψ∆ξ) dx dt. (18.3)

To show (18.3), observe that there exists a sequence of functions ψj ∈ D((0,∞))such that ψj = 0 on (0, t0− 1/j], ∂tψj ≥ 0 on [t0− 1/j, t0], and ψj = ψ on [t0,∞).

Taking ϕ(x, t) := ξ(x)ψj(t) as test-function, it follows that∫ ∞

t0

∫Rn

upξψ dx dt ≤∫ ∞

0

∫Rn

upξψj dx dt = −∫ ∞

0

∫Rn

u(ξ∂tψj + ψj∆ξ) dx dt

≤ −∫ ∞

t0

∫Rn

u(ξ∂tψ + ψ∆ξ) dx dt −∫ t0

t0−1/j

∫Rn

uψj∆ξ dx dt.

Since the last integral goes to 0 by dominated convergence, this yields (18.3).Step 2. Now we take ζ ∈ D(B1) and φ ∈ D((−1, 1)), such that ζ = 1 in B1/2,

φ = 1 in [0, 1/2), and 0 ≤ ζ, φ ≤ 1. Let m = 2p/(p− 1) and define

ξR(x) = ζm( x

R

), x ∈ R

n, ψR(t) = φm( t− t0

R2

), t ≥ t0.

We observe that

∆ξR(x) = mR−2[ζm−1∆ζ + (m− 1)ζm−2|∇ζ|2

]( x

R

)and

∂tψR(t) = mR−2[φm−1φt

]( t− t0R2

),

hence ∣∣ξR∂tψR + ψR∆ξR

∣∣ ≤ CR−2(ξRψR)1/pχR/2<|x|<RχR2/2<t−t0<R2.

Using (18.3) with ξ = ξR, ψ = ψR, and applying Holder’s inequality, we obtain∫ ∞

t0

∫Rn

upξRψR dx dt ≤ CR−2

∫ t0+R2

t0+R2/2

∫R/2<|x|<R

u(ξRψR)1/p dx dt

≤ CR−2+(n+2)(p−1)/p(∫ t0+R2

t0+R2/2

∫R/2<|x|<R

upξRψR dx dt)1/p

.

(18.4)

In particular, it follows that∫ ∞

t0

∫Rn

upξRψR dx dt ≤ CRn+2−2p/(p−1). (18.5)

If p < pF , i.e. n + 2− 2p/(p− 1) < 0, this implies u ≡ 0 upon letting R →∞ andthen t0 → 0. If p = pF , then (18.5) implies

∫∞t0

∫Rn up < ∞. Therefore, the RHS

of (18.4) goes to 0 as R →∞ and we again conclude that u ≡ 0.

The proof of assertion (ii) is postponed to Section 20, where more detailed globalexistence results will be given. Below we present two other proofs of (differentformulations of) the nonexistence part of Theorem 18.1. We shall start with theproof which is due to [529], [530]. Recall that Gt denotes the Gaussian heat kernel,defined in (48.5).

Theorem 18.3. Let 1 0 in a

set of positive measure. Then there is no nonnegative measurable global solutionu : R

n × [0,∞)→ [0,∞] to the integral equation

u(x, t) =∫

Rn

Gt(x− y)u0(y) dy +∫ t

0

∫Rn

Gt−s(x− y)up(y, s) dy ds

and u(x, t) < ∞ for a.e. (x, t) ∈ Rn × (0,∞).

(18.6)

Proof. Assume that there exists a global solution of (18.6). Lemma 15.6 implies

t1/(p−1)Gt ∗ u0 ≤ C. (18.7)

Given a measurable function v : Rn → [0,∞], we have

limt→∞(4πt)n/2Gt ∗ v = ‖v‖1 pointwise in R

n, (18.8)

where ‖v‖1 := ∞ if v /∈ L1(Rn). If p < pF , then (18.7) implies tn/2‖Gt ∗u0‖∞ → 0as t→∞ which contradicts (18.8) with v = u0.

Hence we may assume p = pF . By redefining u on a null set, we may assumethat (18.6) actually holds everywhere in R

n × (0,∞) and it is easy to check that

u(t + t0) = Gt ∗ u(t0) +∫ t

0

Gt−s ∗ up(s + t0) ds, for all t, t0 > 0. (18.9)

We first note that Corollary 15.8 and (18.8) imply the existence of C1 > 0 suchthat

‖u(τ)‖1 ≤ C1, for a.e. τ > 0. (18.10)

On the other hand, since |x− z|2/(4t) ≤ (|x|2 + |z|2)/(2t), we obtain

u(x, t) ≥ (Gt ∗ u0)(x) ≥ (4πt)−n/2e−|x|2/(2t)

∫Rn

e−|z|2/(2t)u0(z) dz.

In particular, we have

u(x, 2) ≥ kG1(x), x ∈ Rn, (18.11)

for some k > 0. Using (18.9) and (48.6), we deduce that

u(s + 2) ≥ Gs ∗ u(2) ≥ kGs ∗G1 = kGs+1, s > 0. (18.12)

Now, Proposition 48.4(a) and (p− 1)n/2 = 1 imply

‖Gps+1‖1 = (4π(s + 1))−(p−1)n/2p−n/2‖G(s+1)/p‖1 = C2(s + 1)−1

for some C2 > 0. This calculation, (18.9) with t0 = 2, (18.12) and Proposi-tion 48.4(b) guarantee

‖u(t + 2)‖1 ≥∫ t

0

‖Gt−s ∗ up(s + 2)‖1 ds ≥∫ t

0

‖Gt−s ∗ (kGs+1)p‖1 ds

= kp

∫ t

0

‖Gps+1‖1 ds = kpC2

∫ t

0

(s + 1)−1 ds →∞

as t→∞, which contradicts (18.10).

The following method is based on the approach of [299] (in Lemma 18.4 below wealso use some ideas from [384]). We introduce the forward similarity variables

y =x√

t + 1, s = log(1 + t)

and define the rescaled function

v(y, s) = eβsu(es/2y, es − 1), β =1

p− 1(18.13)

(in other words, v(y, s) = tβu(x, t)). Problem (18.1) can then be written in theform

vs + Lv = |v|p−1v + βv, y ∈ Rn, s > 0,

v(y, 0) = u0(y), y ∈ Rn,

(18.14)

where

Lv := −∆v − y · ∇v

2= −g−1∇ · (g∇v), g(y) := e|y|

2/4. (18.15)

SetLq

g := f ∈ Lq(Rn) :∫

Rn

|f(y)|qg(y) dy <∞,

H1g := f ∈ L2

g : ∇f ∈ L2g (18.16)

and H2g := f ∈ H1

g : ∇f ∈ H1g. Then

(Lv, w)g = −∫

Rn

∇ · (g∇v)w dy =∫

Rn

(∇v · ∇w)g dy, v, w ∈ H2g ,

where (u, v)g :=∫

Rn uvg dy denotes the scalar product in L2g. Lemmas 47.9, 47.10

and 47.13 show that L is a positive self-adjoint operator in L2g with compact

inverse, domain of definition H2g and eigenvalues λL

k = (n+k−1)/2, k = 1, 2, . . . . Inaddition, φ1(y) := e−|y|2/4 is the eigenfunction corresponding to the first eigenvalueλL

1 .

Denote

E(v) :=∫

Rn

[ |∇v|22

− β

2v2 − 1

p + 1|v|p+1

]g(y) dy.

Notice that E is well defined in H1g if p ≤ pS since H1

g → Lp+1g due to Lemma 47.11.

Let T ∈ (0,∞] and assume that v ∈ C([0, T ), H1g ) is a solution of (18.14). Ex-

ample 51.24 shows that v ∈ C((0, T ), H2g ) ∩ C1((0, T ), L2

g), hence the mappings → E(v(s)) belongs to C([0, T ), R) ∩C1((0, T ), R).

Lemma 18.4. Let 1 < p < pS.(i) The function s → E

(v(s)

)is nonincreasing.

(ii) If β ≤ λL1 and E

(v(s0)

)< 0 for some s0, then T <∞.

(iii) If T = ∞, then there exist positive constants C0 = C0(n, p), C1 = C1(‖u0‖H1g)

and C2 = C2(u0) such that

−C0 ≤ E(v(s)) ≤ C1, (18.17)

‖v(s)‖L2g≤ C1, (18.18)

‖v(s)‖H1g≤ C2 (18.19)

for all s ≥ 0.

Proof. (i) The assertion follows from

d

dsE(v(s)

)= −

∫Rn

v2s (s)g dy ≤ 0.

(ii) Without loss of generality we may assume s0 = 0. Then the proof followsby repeating word-by-word part (ii) of the proof of Theorem 17.6.

(iii) Set Av := Lv − βv, choose ε ∈ (0, (p − 1)/2) and denote c0 :=1 − (2 + 2ε)/(p + 1) > 0. Multiplying the equation vs + Av = |v|p−1v by vgwe obtain

12

d

ds‖v(s)‖2L2

g= −(Av(s), v(s))g + ‖v(s)‖p+1

Lp+1g

= −(2 + 2ε)E(v(s)) + ε(Av(s), v(s))g + c0‖v(s)‖p+1

Lp+1g

.(18.20)

Recall that A is a self-adjoint operator with compact resolvent and its eigenvaluesare λL

k − β, k = 1, 2, . . . . Choose k0 such that λLk0

> β, let P be the spectralprojection in L2

g corresponding to the spectral set λLk0− β, λL

k0+1 − β, . . . andQ = I − P . Notice that

dimQL2g <∞, PL2

g ⊥ QL2g (18.21)

and there exist c1, c2 > 0 such that, for all w ∈ H2g ,

(Aw, w)g = (APw, Pw)g + (AQw, Qw)g

≥ c1‖Pw‖2H1g− c2‖Qw‖2H1

g= c1‖w‖2H1

g− (c1 + c2)‖Qw‖2H1

g.

(18.22)

Fix s > 0. Ifc0‖v(s)‖p+1

Lp+1g

≤ ε(c1 + c2)‖Qv(s)‖2H1g, (18.23)

then Holder’s inequality and (18.21) guarantee the existence of c3, c4 > 0 such that1c3‖Qv(s)‖2H1

g≤ ‖Qv(s)‖2L2

g= (Qv(s), Qv(s))g = (v(s), Qv(s))g

≤ ‖v(s)‖Lp+1g‖Qv(s)‖

L(p+1)′g

≤ c4‖Qv(s)‖2/(p+1)H1

g‖Qv(s)‖H1

g,

hence ‖Qv(s)‖H1g≤ c5 and (18.20), (18.22) guarantee

12

d

ds‖v(s)‖2L2

g≥ −(2 + 2ε)E(v(s)) + c1‖v(s)‖2H1

g− c5 (18.24)

for some c1, c5 > 0. If (18.23) fails, then (18.20), (18.22) guarantee (18.24) withc5 = 0.

Assume s0 ≥ 0 and

−(2 + 2ε)E(v(s0)) > c5 + 1 or c1‖v(s0)‖2L2g

> c5 + 1 + (2 + 2ε)E(u0). (18.25)

Then (18.24), the inequality E(v(s0)) ≤ E(u0) and the identity

E(v(s)) = E(v(s0))−∫ s

s0

‖vs(t)‖2L2gdt, s ≥ s0, (18.26)

imply12

d

ds‖v(s)‖2L2

g≥ (2 + 2ε)

∫ s

s0

‖vs(t)‖2L2gdt + 1, s ≥ s0.

Set f(s) := 12

∫ s

s0‖v(t)‖2L2

gdt. Then the same arguments as in the proof of Theo-

rem 17.6 show that the function s → f(s)−ε is concave for s large which contradictsthe assumption T = ∞. Consequently, (18.25) fails and (18.17), (18.18) are true.

Notice that (18.26) and (18.17) imply∫ ∞

0

‖vs(s)‖2L2gds ≤ C1 + C0 (18.27)

and (18.24), (18.17), (18.18) and Cauchy’s inequality guarantee the existence ofc6, c7 > 0 such that

‖vs(s)‖2L2g≥ c6‖v(s)‖4H1

g− c7. (18.28)

Set Λt := s ≥ t : c6‖v(s)‖4H1g

> c7 + 1 and let |Λt| denote the measure ofΛt. Then |Λt| → 0 as t → ∞ due to (18.27) and (18.28). The well-posedness of(18.14) in H1

g (see Example 51.24) guarantees the existence of η, c8 > 0 such that‖v(s + τ)‖H1

g≤ c8 whenever τ ∈ [0, η] and s /∈ Λ0. Fix t > 0 such that |Λt| < η.

Then ‖v(s)‖H1g≤ c8 for all s ≥ t + η, which proves (18.19).

Remark 18.5. The constant C2 in (18.19) depends on ‖u0‖H1g

only. In fact,let Λt be the set in the proof of Lemma 18.4(iii). Since |Λ0| < C1 + C0 due to(18.27) and (18.28), in any interval of the form [s, s + C1 + C0], s > 0 we canfind s0 such that ‖v(s0)‖H1

g≤ C3 for some C3 = C3(‖u0‖H1

g) (and the same

is true for all s0 > 0 close to zero). Due to the smoothing estimates in Exam-ple 51.24 we may also assume ‖v(s0)‖∞ ≤ C3. Now estimate (5.26) in [440, Theo-rem 5.3] (used with u(x, t) = (t + 1)−1/(p−1)v(x/

√t + 1, log(t + 1) + s0)) guaran-

tees ‖v(s)‖∞ ≤ C4 for some C4 = C4(‖u0‖H1g) and all s ∈ [s0, s0 + 2(C0 + C1)].

Consequently, ||v|p−1v| ≤ Cp−14 |v| and an easy estimate based on the variation-of-

constants formula guarantees ‖v(s)‖H1g≤ C5 for some C5 = C5(‖u0‖H1

g) and all

s ∈ [s0, s0 + 2(C0 + C1)].

Another proof of Theorem 18.1(i) for classical solutions. Let p ≤ pF ,0 ≤ u0 ∈ L∞(Rn), u0 ≡ 0, and assume that the corresponding maximal classicalsolution u of (18.1) is global. Similarly as in the proof of Theorem 18.3, we have(18.11), hence u(·, 2) ≥ c0φ1 for some c0 > 0. Due to the maximum principle, thesolution v of (18.14) starting at v0 := c0φ1 exists globally.

First assume p < pF . Since the solution v is global, Lemma 18.4(iii) guaranteesthat it is bounded in H1

g . On the other hand,

v(t) ≥ e−t(L−β)(c0φ1) = c0et(β−λL

1 )φ1

and β − λL1 > 0, which yields a contradiction.

Now assume p = pF . Using (Lφ1, φ1)g = λL1 (φ1, φ1)g and β = λL

1 we obtain

E(c0φ1) = − cp+10

p + 1

∫Rn

φp+11 (y)g(y) dy < 0,

which contradicts Lemma 18.4(ii).

Remarks 18.6. (i) Alternative proof. In [299], another contradiction argumentwas used in the case p < pF : Let v be the global solution starting at v0 := c0φ1.Set ψ := bεφ

1+ε1 where ε > 0 and bε > 0 is such that

∫Rn ψg dy = 1. Notice that

Lψ = (1 + ε)λL1 ψ − ε(1 + ε)bεφ

ε−11 |∇φ1|2 ≤ (1 + ε)λL

1 ψ

and set f(s) := (v(s), ψ)g . Then Jensen’s inequality implies

d

dsf(s) =

∫Rn

v(y, s)pψ(y)g(y) dy + β(v(s), ψ

)g−(v(s), Lψ

)g

≥(∫

Rn

v(y, s)ψ(y)g(y) dy)p

+[β − (1 + ε)λL

1

](v(s), ψ

)g

= f(s)p +[β − (1 + ε)λL

1

]f(s).

Due to p < pF there exists ε > 0 such that β = 1/(p−1) ≥ (1+ε)n/2 = (1+ε)λL1 ,

hence f ′ ≥ fp, f(0) > 0, which contradicts the global existence of f .(ii) Other domains. Consider problem (15.1) in the half-space Ω = R

n+ :=

x ∈ Rn : xn > 0. Repeating the last proof of Theorem 18.3 we obtain a self-

adjoint operator L with the first eigenvalue λL1 = (n+1)/2 and the corresponding

eigenfunction φ1(y) = yne−|y|2/4. Consequently, the problem does not possessnontrivial nonnegative global solutions if p ≤ 1 + 2/(n + 1). Of course, instead ofGt one has to work with the kernel Gt(x, z) = Gt(x − z)

(1 − e−xnzn/t

). If Ω =

(0,∞)n, then analogous arguments show nonexistence of global positive solutionsfor p ≤ 1 + 1/n (the first eigenfunction is y1y2 . . . yne−|y|2/4).

(iii) A characterization of the critical exponent. It has been observedin [363] that, for any domain Ω, the critical Fujita exponent pF = pF (Ω) canbe characterized in terms of maximal decay rate of the heat semigroup. Namely,denoting

a∗ := sup

a > 0 : supt∈(0,∞)

ta‖e−tAu0‖∞ <∞ for some 0 ≤ u0 ∈ L∞(Ω), u0 ≡ 0

,

(18.29)there holds

pF = 1 +1a∗ .

Indeed, if 1 1 + (1/a∗), by takingu0 such that a in (18.29) satisfies 1/(p− 1) < a < a∗, we deduce from the proof ofTheorem 20.2 below (with R

n replaced by Ω) that Tmax(u0) = ∞.(iv) Sign-changing solutions. Consider problem (18.1) with n = 1 and set

Λk = u : u has exactly k sign changes.

Then there exists a global solution of (18.1) with u0 ∈ Λk if and only if p >1 + 2/(k + 1). In addition, if p > 1 + 2/(k + 1), then there exists a global solutionof (18.1) with u0 ∈ Λk ∩ H1

g (see [384] and [385]). Notice that 1 + 2/(k + 1) =1 + 1/λL

k+1.

We close this section with an application of Theorem 18.1 to a model arisingin population genetics [211], [35]. In that model, a biological species possesses agene existing in two allelic forms A and a, leading to the three genotypes AA,Aa and aa. It is assumed that the death rate of the individuals is determined bythis particular gene, and the death rates corresponding to the genotypes AA, Aa,aa are respectively denoted by k1, k2, k3. Moreover, it is assumed that k2 = k3

and that the genotype AA is advantageous in the sense that k1 < k2. Denote byu : R

n × [0,∞) → [0, 1] the relative density of the gene A at point x and time t,and set k = k2 − k1 > 0. Under suitable physical assumptions, the equation for uis then given by

ut −∆u = ku2(1− u), x ∈ Rn, t > 0. (18.30)

This equation is supplemented with the initial condition

u(x, 0) = u0(x), x ∈ Rn, (18.31)

where u0 ∈ X := φ ∈ C(Rn) : 0 ≤ φ(x) ≤ 1, x ∈ Rn. It follows from Re-

mark 51.11 and the comparison principle that problem (18.30)–(18.31) admits aunique, global, classical solution u and that 0 ≤ u(x, t) ≤ 1 in R

n × (0,∞).We have the following result [35] concerning the asymptotic behavior of solutions

(our proof is a simplification of arguments in [35]).

Theorem 18.7. Consider problem (18.30)–(18.31).(i) If n = 1 or 2, then u = 1 is globally stable in the following sense: For anyu0 ∈ X, u0 ≡ 0, there holds

limt→∞u(x, t) = 1,

uniformly on compact subsets.(ii) If n ≥ 3, then there exist positive u0 ∈ X such that

limt→∞ ‖u(t)‖∞ = 0.

Remarks 18.8. (i) The phenomenon displayed in Theorem 18.7(i) is called the“hair-trigger effect”: Any small perturbation from the rest-state u ≡ 0 drives thesolution to the equilibrium u ≡ 1, leading to the eventual extinction of the gene a.

(ii) Equation (18.30) is a special case of a more general class of equations of theform ut−∆u = f(u), where the nonlinearity satisfies f(0) = f(1) = 0, which arisein various biological models and also in flame propagation models from combustiontheory. An important case is the so-called Fisher-KPP equation, corresponding tof(u) = u(1 − u). Starting with the pioneering works [211], [308], a very largeamount of literature has been devoted to these problems, in particular to theexistence of traveling wave solutions and to their analysis. These are solutions ofthe form u(x, t) = w(x1 − ct), connecting the equilibria u ≡ 0 and u ≡ 1 (i.e.w(−∞) = 1, w(+∞) = 0). See [35], [179], and e.g. [266] and the references thereinfor more recent results.

(iii) Simple modifications of the proof of Theorem 18.7 show the following.Assume that the nonlinearity in (18.1) is replaced by any C1 function f : [0,∞)→[0,∞) such that

f(u) ≥ kup, u ∈ [0, b], for some k, b > 0 and 1 ≤ p ≤ pF . (18.32)

Then any positive solution is either nonglobal or satisfies lim inft→∞ u(x, t) ≥ b,uniformly on compact subsets. In that sense, the Fujita-type result can be seen asan instability property of u = 0 for small perturbations, which essentially dependson the behavior of f for small u. This explains the fact that the instable rangecorresponds to small p’s. On the other hand, if in addition to (18.32) we assumethat f is convex and satisfies the blow-up condition (17.4), then it is easy to showthat any positive solution is nonglobal.

For the proof of assertion (i), we need the following lemma.

Lemma 18.9. For all ε > 0, there exist Rε > 0 and a function φε ∈ C2(BRε)such that

0 ≤ φε(x) ≤ 1− ε, x ∈ BRε

andvε(0, t) ≥ 1− ε, t ≥ 0, (18.33)

where vε is the solution of the problem

vt −∆v = v2(1− v), x ∈ BRε , t > 0,

v = 0, x ∈ ∂BRε , t > 0,

v(x, 0) = φε(x), x ∈ BRε .

⎫⎪⎬⎪⎭Proof. Assume ε ∈ (0, 1/2) without loss of generality. Fix a nontrivial nonneg-ative radial function h ∈ D(Rn) such that h(x) = 0 for |x| ≥ 1/2. Let ϕ be theclassical solution of

−∆ϕ = h, |x| < 1,

ϕ = 0, |x| = 1,

and observe that ϕ is positive, radial nonincreasing. Let

φ(x) = φε(x) := (1− ε)ϕ(x/R)

ϕ(0)≤ 1− ε, |x| ≤ R,

where R > 0 is to be fixed. For |x| ≤ R/2, we have φ ≥ c := (2ϕ(0))−1ϕ(1/2) > 0,hence

∆φ + φ2(1− φ) ≥ ∆φ + εc2 ≥ −(ϕ(0))−1R−2‖∆ϕ‖∞ + εc2 > 0

provided we take R = Rε > 0 large enough. Since ∆φ = 0 for |x| ≥ R/2, we obtain

∆φ + φ2(1 − φ) ≥ 0, x ∈ BRε .


It follows from Proposition 52.19 that ∂tvε ≥ 0, hence in particular

vε(0, t) ≥ φ(0) = 1− ε, t ≥ 0.

Proof of Theorem 18.7(i). We may assume k = 1 without loss of generality.Step 1. Let v0 ∈ X , v0 ≡ 0, be such that v0(x0 + ·) is radial nonincreasing

for some x0 ∈ Rn, and let v be the solution of (18.30) with initial data v0. Then

v(x + x0, t) is also radial nonincreasing (cf. Proposition 52.17). We claim that

lim supt→∞

v(x0, t) = 1.

Assume the contrary. Then there exist ε ∈ (0, 1) and T > 0 such that v(x, t) ≤1−ε in R

n× [T,∞). Consequently, w := εv satisfies wt−∆w ≥ w2 in Rn× [T,∞).

Since 2 ≤ pF due to n ≤ 2, it follows from Theorem 18.1(i) and Remark 18.2(i)that w is nonglobal: a contradiction.

Step 2. Let u0 ∈ X , u0 ≡ 0. We claim that for all ε, R > 0, there exists t0 > 0such that

u(x, t0) ≥ 1− ε, |x| ≤ R. (18.34)

By a time shift, we may assume without loss of generality that u0 > 0 in Rn. There-

fore, for any x0 ∈ Rn, u0 dominates some nontrivial v0 ∈ X such that v0(x0 + ·)

is radial nonincreasing. If follows from Step 1 and the comparison principle thatfor all x0 ∈ R

n,lim sup

t→∞u(x0, t) = 1.

If u0 is radial nonincreasing, then this readily implies (18.34). The general casefollows from the fact that u0 dominates some nontrivial, radial nonincreasing v0 ∈X .

Step 3. Let u0 ∈ X , u0 ≡ 0. Fix ε ∈ (0, 1) and M > 0. Let Rε, φε be given byLemma 18.9. By Step 2, applied with R = Rε + M , there exists t0 > 0 such that

u(x0 + x, t0) ≥ 1− ε ≥ φε(x), |x| ≤ Rε, |x0| ≤M.

By the comparison principle and (18.33), we conclude that

u(x0, t) ≥ vε(0, t) ≥ 1− ε, |x0| ≤M, t ≥ t0.

The assertion is proved.(ii) Since 2 > pF due to n ≥ 3, this is an immediate consequence of Theo-

rem 18.1(ii) and of the comparison principle.

19. Global existence for the Dirichlet problem

19.1. Small data global solutions

We start with a basic result of global existence for small initial data for the problem

ut −∆u = f(u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (19.1)

Definition 19.1. Assume that f(0) = 0 (so that u ≡ 0 is a solution to (19.1))and that (19.1) is locally well-posed in a space X . We say that the zero solutionis asymptotically stable in X if there exists a constant η > 0 such that, for allu0 ∈ X with ‖u0‖X ≤ η, there holds Tmax(u0) = ∞ and

limt→∞ ‖u(t)‖X = 0.

We say that the zero solution is exponentially asymptotically stable in X ifthere exist constants η, µ > 0 and K ≥ 1 such that, for all u0 ∈ X with ‖u0‖X ≤ η,there holds Tmax(u0) = ∞ and

‖u(t)‖X ≤ K‖u0‖Xe−µt, t > 0.

Theorem 19.2. Consider problem (19.1), where Ω is bounded and f : R → R

is a C1-function such that f(0) = 0 and f ′(0) < λ1. Then the zero solution isexponentially asymptotically stable in L∞(Ω).

Theorem 19.2 can be given a simple proof based on the comparison principle(see [296] for similar arguments).

Proof. By assumption, there exist η > 0, ε ∈ (0, λ1/2) such that

|f(s)| ≤ (λ1 − 2ε)|s|, |s| ≤ η. (19.2)

We claim that there exists a function ϕ ∈ C2(Ω) such that

−∆ϕ = (λ1 − ε)ϕ and ϕ ≥ 1, x ∈ Ω. (19.3)

Indeed, it suffices to consider ϕ = 1 + ψ, where ψ is the solution of

−∆ψ = (λ1 − ε)ψ + (λ1 − ε), x ∈ Ω,

ψ = 0, x ∈ ∂Ω,

19. Global existence for the Dirichlet problem 113

and to note that ψ ≥ 0 by the maximum principle. Next set

u(x, t) = η e−εtϕ(x), where η =(max

Ωϕ)−1

η.

An obvious computation and (19.3), (19.2) yield

ut −∆u = (λ1 − 2ε)u ≥ f(u), x ∈ Ω, t > 0.

Assume that ‖u0‖∞ ≤ η, hence |u0| ≤ u(·, 0). By the comparison principle, wededuce that u ≤ u in Ω × (0, Tmax(u0)), and we get u ≥ −u by arguing on −u.The conclusion follows.

Let us now consider in more detail the case of the model problem (15.1). Wewould like to extend Theorem 19.2 in two directions:

• unbounded domains;

• Lq- instead of L∞-stability. Note that this is a legitimate question for q > qc :=n(p − 1)/2 or q = qc > 1, since we know (cf. Theorems 15.2 and 15.3 andRemark 15.4) that problem (15.1) is locally well-posed in Lq(Ω) for (and onlyfor) such q.

Domains that admit such extension can be characterized geometrically throughthe notion of inradius. Recall (see Appendix D) that the inradius of Ω is definedby:

ρ(Ω) = supr > 0 : Ω contains a ball of radius r

= sup

x∈Ωdist(x, ∂Ω)

and that, for any q ∈ [1,∞], the condition ρ(Ω) <∞ is equivalent to the Poincareinequality

‖φ‖q ≤ C(Ω, q)‖∇φ‖q, φ ∈ W 1,q0 (Ω) (19.4)

(provided Ω is uniformly smooth).

The following result of [481], [483] asserts in particular that for any qc < q ≤ ∞,the zero solution is asymptotically stable in Lq(Ω) if and only if Ω has finiteinradius.

Theorem 19.3. Consider problem (15.1) with p > 1 and let 1 ≤ q ≤ ∞.

(i) Assume q > qc or q = qc > 1. If ρ(Ω) < ∞, then the zero solution is exponen-tially asymptotically stable in Lq(Ω).

(ii) Assume q > qc. If ρ(Ω) =∞, then the zero solution is not asymptotically stablein Lq(Ω). More precisely, there exist initial data u0 ∈ Lq(Ω) of arbitrarily smallLq-norm such that Tmax(u0) < ∞.

Remarks 19.4. (a) Critical case. The result of Theorem 19.3(ii) is no longertrue in the critical case q = qc: We shall see later that for any domain Ω (includ-ing the whole space), the zero solution is asymptotically stable in Lqc(Ω) — seeCorollary 20.20 and Remark 20.21. However, the stability is exponential only ifρ(Ω) <∞ (see [481]).

(b) Different methods of proof. Theorem 19.3(i) for 1 < q < ∞ can beproved by a multiplier argument, using multiplication by powers of u and thePoincare inequality [481]. We shall employ this method here, but for simplicity weshall prove the result only in the range 2 ≤ q < ∞ (the idea for 1 < q < 2 is thesame, but some additional technical difficulties arise).

An alternative proof, covering the extremal cases q = 1 and q = ∞ as well,can be carried out by using the variation-of-constants formula and the exponentialdecay of the heat semigroup for ρ(Ω) <∞ (see [483]). Such arguments can be usedto prove more general results of linearized stability; see Theorems 51.17, 51.19 and51.33 in Appendix E.

As an advantage, the energy proof might also apply to certain quasilinear prob-lems.

Proof of Theorem 19.3(i) for 2 ≤ q < ∞. To simplify notation, if k is anypositive number, we write uk for sign(u) |u|k. Since u0 ∈ Lq(Ω), it follows from Ex-ample 51.27 in Appendix E that u ∈ C([0,∞), Lq(Ω))∩C((0,∞), W 2,q∩W 1,q

0 (Ω))∩C1((0,∞), Lq(Ω)). Multiplying the equation by uq−1 and integrating by parts, weobtain

1q

d

dt‖u(t)‖q

q = 〈uq−1, ∆u〉+ ‖u‖q+p−1q+p−1 = −4(q − 1)

q2‖∇(uq/2)‖22 + ‖u‖q+p−1

q+p−1,

(19.5)for all t ∈ (0, Tmax(u0)). For the last term of inequality (19.5), we next establishthe estimate

‖u‖q+p−1 ≤ C‖u‖θq‖∇(uq/2)‖2(1−θ)/q

2 , (19.6)

with

θ = 1− n(p− 1)2(q + p− 1)

∈ (0, 1). (19.7)

To do so, let us consider separately the cases n ≥ 3 and n ≤ 2. If n ≥ 3, sinceq ≥ n(p− 1)/2, we have q + p− 1 ≤ nq/(n− 2) hence, by Holder’s inequality,

‖u‖q+p−1 ≤ ‖u‖θq ‖u‖1−θ

nq/(n−2),

with

θ =(

1q + p− 1

− 1nq/(n− 2)

)(1q− 1

nq/(n− 2)

)−1

= 1− n(p− 1)2(q + p− 1)

,

and Sobolev’s inequality then yields (19.6).

If n ≤ 2, we use the Gagliardo-Nirenberg inequalities

‖v‖a ≤ Ca‖v‖(a+2)/2a2 ‖v′‖(a−2)/2a

2 , a ≥ 2, v ∈ H10 (Ω) (n = 1)

and‖v‖a ≤ Ca‖v‖2/a

2 ‖∇v‖1−(2/a)2 , a ≥ 2, v ∈ H1

0 (Ω) (n = 2).

Applying this with v = uq/2(t) and a = 2(q + p − 1)/q > 2 yields (19.6) withθ = q/(q + p− 1) if n = 2 and θ = (2q + p− 1)/2(q+ p− 1) if n = 1, that is (19.7).

The next step is to use the Poincare inequality (50.2) in W 1,q0 (Ω) (valid due

to ρ(Ω) < ∞; see Proposition 50.1 in Appendix D) to obtain a lower estimate ofthe first term in the right-hand side of (19.5). It follows from (50.2) that, for allα ∈ [0, 1],

‖∇(uq/2)‖22 ≥ C‖u‖qq + C‖∇(uq/2)‖2α

2 ‖∇(uq/2)‖2(1−α)2

≥ C‖u‖qq + C‖u‖qα

q ‖∇(uq/2)‖2(1−α)2 .

(19.8)

On the other hand, one has (1− θ)(q + p− 1)/q = n(p− 1)/2q ≤ 1. Therefore, wemay choose

α = 1− (1− θ)(q + p− 1)/q,

and by combining (19.5), (19.6) and (19.8), it follows that

1q

d

dt‖u(t)‖q

q ≤ −C‖u‖qq + C‖∇(uq/2)‖2(1−α)

2 ‖u‖qαq

(‖u‖p−1

q − C′).It follows from this differential inequality that if ‖u0‖q is sufficiently small, thenfor all t > 0,

d

dt‖u(t)‖q

q ≤ −C ‖u‖qq,

hence‖u(t)‖q ≤ e−C′t‖u0‖q, (19.9)

as long as the solution exists. If q > qc, we know from Theorem 15.2 that theLq-norm must blow up if Tmax(u0) is finite. The estimate (19.9) thus ensuresglobal existence. If q = qc, global existence when ‖u0‖q is small follows fromCorollary 20.20 below.

Proof of Theorem 19.3(ii). Fix a test-function ϕ ∈ D(Rn), ϕ ≥ 0, ϕ ≡ 0with supp(ϕ) ⊂ B := B(0, 1), and let w be the solution of problem (15.1) with Ωreplaced by B and u0 replaced by ϕ. Due to e.g. Theorem 17.1, we can assumethat w blows up in a finite time T (replacing ϕ by a sufficiently large multiple).

Now, since ρ(Ω) = ∞, Ω contains some ball Bk = B(xk, k) for any integerk ≥ 1. Let us set

uk(x, t) := k−2/(p−1)w(k−1(x− xk), k−2t), u0,k(x) := k−2/(p−1)ϕ(k−1(x− xk)).

Due to the invariance of the equation under this scaling, it is easily verified thatuk solves the problem

∂tuk −∆uk = |uk|p−1uk, x ∈ Bk, 0 < t < k2T,

uk = 0, x ∈ ∂Bk, 0 < t < k2T,

uk(x, 0) = u0,k(x), x ∈ Bk.

⎫⎪⎬⎪⎭Let uk be the solution of problem (15.1) with u0 = u0,k. Since each Bk is includedin Ω and uk ≥ 0 on ∂Bk, it follows from the comparison principle that uk ≥ uk,hence uk blows up in finite time.

Last, an easy calculation yields

‖u0,k‖q = k−2/(p−1)+n/q‖ϕ‖q → 0, k →∞,


We shall now describe the potential well method. It will enable us to obtainalternative sufficient conditions for global existence (and nonexistence) for themodel problem (15.1).

In the rest of this subsection we assume Ω bounded and 1 < p ≤ pS . Recall thatthe energy functional E is given by

E(u) =12

∫Ω

|∇u|2 dx− 1p + 1

∫Ω

|u|p+1 dx, u ∈ H10 (Ω). (19.10)

We define the Nehari functional I by

I(u) =∫

Ω

|∇u|2 dx−∫

Ω

|u|p+1 dx, u ∈ H10 (Ω).

The potential well associated with problem (15.1) is the set

W :=u ∈ H1

0 (Ω) : E(u) < d, I(u) > 0∪ 0,

where d, the depth of the potential well, is defined by

d := infE(u) : u ∈ H1

0 (Ω) \ 0, I(u) = 0. (19.11)

We shall show in Lemma 19.7(i) below that

d =p− 1

2(p + 1)Λ2(p+1)/(p−1), (19.12)

where Λ = Λp+1(Ω) denotes the best constant in the Sobolev embedding H10 (Ω) →

Lp+1(Ω), i.e.,

Λ := inf‖∇u‖2‖u‖p+1

: u ∈ H10 (Ω), u = 0

. (19.13)

The exterior of the potential well is the set

Z :=u ∈ H1

0 (Ω) : E(u) < d, I(u) < 0.

In what follows, for u0 ∈ H10 (Ω), u denotes the maximal Lp+1-classical solution of

problem (15.1) (recall from Section 15 that (15.1) is well-posed in Lp+1(Ω), sincep + 1 ≥ qc due to p ≤ pS).

Theorem 19.5. Consider problem (15.1) with Ω bounded.(i) Assume 1 0,

and‖u(t)‖∞ → 0, t→∞. (19.14)

(ii) Assume 1 < p ≤ pS. If u0 ∈ Z, then Tmax(u0) <∞.

The potential well method was introduced in [467] to obtain global existenceresults for nonlinear hyperbolic equations. As for parabolic problems, the globalexistence part in Theorem 19.5(i) is due to [515] and the decay property is es-sentially from [290]. Theorem 19.5(ii) is due to [409] (see also [290]), where thepotential well method was extended to obtain nonexistence results for hyperbolicand parabolic problems.

Remarks 19.6. (a) Theorem 19.5(i) provides in particular a sufficient smallnesscondition on u0 for global existence when p < pS . Indeed, we have u0 ∈ Wwhenever u0 ∈ H1

0 (Ω) satisfies ‖∇u0‖ <√

2d (cf. Lemma 19.7(iii)).(b) The quantity d can be interpreted as a mountain-pass energy (cf. Section 7).

Indeed, for p ≤ pS , it is easy to show that

d = infu∈H1

0 (Ω)\0maxs≥0

E(su).

Note that for p < pS , there exist least-energy stationary solutions v, i.e.: such thatE(v) = d (this follows from Theorem 7.2, applied with u0 = 0 and u1 such thatE(u1) < 0, and from the easy fact that d = β, where β is defined in (7.1)).

(c) The sets W and Z are invariant under the semiflow associated with problem(15.1) for p ≤ pS. This follows from the proof of Theorem 19.5.

(d) Theorem 19.5 admits a converse (cf. [287]). Namely, if p ≤ pS and u is aglobal solution satisfying (19.14), then u(t) ∈ W for large t. If p < pS and u is a

blowing-up solution, then u(t) ∈ Z for t close to Tmax(u0). These facts respectivelyfollow from smoothing effects and Theorem 19.5(ii), and from property (22.28) inProposition 22.11 and I(u) ≤ 2E(u).

(e) Theorems 17.6 and 19.5 give an essentially complete characterization ofglobal existence/nonexistence in the subcritical range for initial data with energyless than d. See [236] and the references therein for additional information, includ-ing some partial results for higher energy data. It seems an open question whetheror not I(u0) < 0 is a sufficient condition for blow-up.

In view of the proof of Theorem 19.5, we need the following properties of thepotential well.

Lemma 19.7. Let Ω be bounded and let 1 0, there holds

dε := infE(u) : u ∈ H1

0 (Ω), I(u) = −ε≥ d− ε

p + 1. (19.15)

(iii) For all u ∈ H10 (Ω), we have

‖∇u‖2 <√

2d =⇒ u ∈ W =⇒ ‖∇u‖2 <√

2(p+1)p−1 d . (19.16)

Proof. Denote D = p−12(p+1)Λ

2(p+1)/(p−1) and fix ε ≥ 0. Let u ∈ H10 (Ω) satisfy

I(u) = −ε, and assume in addition that u = 0 if ε = 0. Then

E(u) =p− 1

2(p + 1)

∫Ω

|∇u|2 dx− ε

p + 1. (19.17)

Since, by (19.13),∫Ω

|∇u|2 dx ≤∫

Ω

|u|p+1 dx ≤ Λ−(p+1)(∫

Ω

|∇u|2 dx)(p+1)/2

and u = 0, we get∫Ω|∇u|2 dx ≥ Λ2(p+1)/(p−1). This combined with (19.17) implies

d ≥ D anddε ≥ D − (p + 1)−1ε, ε > 0. (19.18)

Let now uj be a minimizing sequence for (19.13). By multiplying uj with suitableµj > 0, we may assume that I(uj) = 0. Therefore∫

Ω

|∇uj |2 dx =∫

Ω

|uj|p+1 dx = (Λ + ηj)−(p+1)(∫

Ω

|∇uj |2 dx)(p+1)/2

,

where ηj → 0+. Combining this with (19.17) for ε = 0, we obtain

E(uj) =p− 1

2(p + 1)(Λ + ηj)2(p+1)/(p−1) → D,

hence d = D, i.e. (19.12). If p < pS , then the infimum in (19.13) is attained forsome v ∈ H1

0 (Ω), due to the compactness of the embedding H10 (Ω) → Lp+1(Ω).

Arguing similarly as above, with uj replaced by v, we see that the infimum in(19.11) is also attained.

Assertion (ii) follows from (19.18).

Finally, let us prove assertion (iii). Assume 0 < ‖∇u‖2 <√

2d. Then E(u) < d.Next, using (19.13) and ‖∇u‖2 <

√2d < Λ(p+1)/(p−1), we obtain∫

Ω

|u|p+1 dx ≤ Λ−(p+1)(∫

Ω

|∇u|2 dx)(p+1)/2

<

∫Ω

|∇u|2 dx.

Consequently I(u) > 0, hence u ∈ W .

On the other hand, for any u ∈ W , the conditions E(u) < d and I(u) ≥ 0 imply

p− 12(p + 1)

∫Ω

|∇u|2 dx ≤ E(u) < d,

hence the last inequality in (19.16).

Proof of Theorem 19.5. Set T := Tmax(u0). By (17.7), we have

E(t) ≤ E(u0) < d, t ∈ [0, T ). (19.19)

(i) If u(t) = 0 for some t ≥ 0, then by uniqueness, u(s) = 0 for all s ≥ t andthe conclusion is true. Hence we may assume that u(t) = 0 for all t ∈ [0, T ).Since I(u0) > 0, using (19.11) and (19.19), it follows by continuity that, for allt ∈ [0, T ), I(u(t)) > 0, hence u(t) ∈W . By Lemma 19.7(iii), we deduce that u(t) isbounded in H1

0 (Ω), hence in Lp+1(Ω). Remarks 16.2 then guarantee that T = ∞.On the other hand, by Example 53.7 (and in particular the existence of a strictLyapunov functional given by (19.10)), the ω-limit set ω(u0) in the H1

0 (Ω)-topologyis nonempty and consists of (classical) equilibria. But for any nontrivial equilibriumv, we have I(v) = 0, hence E(v) ≥ d by (19.11). Consequently v ∈ ω(u0) in viewof (19.19). In other words, limt→∞ ‖u(t)‖1,2 = 0, hence limt→∞ ‖u(t)‖p+1 = 0. Bythe smoothing estimate (15.2), this guarantees (19.14).

(ii) Fix ε > 0 such that

ε < min(−I(u0), d− E(u0)

).

By (19.15) and (19.19), we have E(t) ≤ E(u0) < dε for t ∈ [0, T ). Since I(u0) < −ε,using the definition of dε in (19.15), it follows by continuity that I(u(t)) < −ε forall t ∈ [0, T ), hence

12

d

dt

∫Ω

u2 dx = −I(u(t)) > ε (19.20)

(cf. (17.9)). But on the other hand, we know from Remark 17.7(ii) that T = ∞implies supt≥0 ‖u(t)‖2 < ∞. In view of (19.20), we conclude that T <∞.

19.2. Structure of global solutions in bounded domains

In this subsection we study some properties of the set of initial data giving riseto global solutions of problem (19.1). Throughout this subsection we assume thatthe domain Ω is bounded. We define the sets

G =u0 ∈ L∞(Ω) : Tmax(u0) = ∞

,

B =u0 ∈ L∞(Ω) : Tmax(u0) =∞ and sup

t≥0‖u(t)‖∞ < ∞

,

andD =

u0 ∈ L∞(Ω) : Tmax(u0) = ∞ and ‖u(t)‖∞ → 0, t→∞

(D is the domain of attraction of 0). When (19.1) admits both global andnonglobal solutions, these sets are natural and interesting objects. Clearly, D ⊂B ⊂ G.

In order to describe the properties of these sets, we first need some propertiesof the steady states of (19.1), i.e. (classical) solutions of

−∆u = f(u), x ∈ Ω,

u = 0, x ∈ ∂Ω.

(19.21)

The following result implies in particular that two ordered positive steady statescannot exist when the nonlinearity is strictly convex. Note that the result failsin general if Ω = R

n (with u, v → 0 at infinity), as shown by Theorem 9.1 withp ≥ pJL.

Proposition 19.8. Assume Ω bounded and let f : R → R be a strictly convexC1-function, with f(0) = 0. Assume that u, v ∈ C2(Ω) ∩ C1(Ω) are respectivelysub- and supersolutions to (19.21), in the sense that

−∆u ≤ f(u), x ∈ Ω,

−∆v ≥ f(v), x ∈ Ω,

u = v = 0, x ∈ ∂Ω.

⎫⎪⎬⎪⎭ (19.22)


If v ≥ u > 0 in Ω, then u ≡ v.

Proof. Multiplying the inequalities in (19.22) by v and u respectively, we obtain∫Ω

f(u)v dx ≥∫

Ω

(−∆u)v dx =∫

Ω

∇u · ∇v dx =∫

Ω

(−∆v)u dx ≥∫

Ω

f(v)udx,

hence ∫Ω

(f(u)u

− f(v)v

)uv dx ≥ 0.

But in view of the strict convexity of f , the integrand is nonpositive in Ω, and(strictly) negative at each x such that v(x) > u(x). The conclusion follows.

The next result describes some basic geometrical and topological properties ofthe sets D,B,G. Here we refer to the L∞-topology (but other choices are possible).Also, for a given convex subset K of a vector space, we recall that x ∈ K is calledan extremal point if it cannot be written under the form x = θy + (1− θ)z withy, z ∈ K and θ ∈ (0, 1). Theorem 19.9 is due to [342]. However the present proofof assertion (iv) is different and simpler than that in [342].

Theorem 19.9. Consider problem (19.1) where Ω is bounded and f : R → R is aC1-function, with f(0) = f ′(0) = 0.(i) Then D is an open neighborhood of 0.

(ii) Assume that f is convex. Then the sets G,B and D are convex.

Now assume that f is strictly convex.(iii) If u0 is not an extremal point of G (resp., of B), then u0 is an interior point.This is true in particular if 0 ≤ u0 ≤ v0, with v0 ∈ G (resp., B), v0 ≡ u0.(iv) There holds int(B) = D.

Proof. (i) This is a consequence of Theorem 19.2 and of the continuous depen-dence of solutions on initial values.

(ii) Let θ ∈ (0, 1), u0, v0 ∈ L∞(Ω), u0 ≡ v0, w0 = θu0 +(1−θ)v0, and denote byu, v, w the solutions of (19.1) with initial data u0, v0, w0, respectively. Set w = θu+(1− θ)v. By the convexity of f , for all x ∈ Ω and t ∈ (0, min(Tmax(u0), Tmax(v0))),we have

wt −∆w = θf(u) + (1 − θ)f(v) ≥ f(θu + (1− θ)v) = f(w), (19.23)

hencew ≤ θu + (1 − θ)v, (19.24)

in view of the comparison principle. On the other hand, the assumptions on f implyf(s) ≥ 0, s ∈ R. Denoting by e−tA the heat semigroup in Ω with homogeneous

Dirichlet boundary conditions, the maximum principle and Proposition 48.5 inAppendix B imply

w ≥ e−tAw0 ≥ −Ce−λ1t. (19.25)

It then follows from (19.24) and (19.25) that w0 ∈ G (resp., B, D) wheneveru0, v0 ∈ G (resp., B, D) and the convexity assertion is proved.

Assume now that f is strictly convex.(iii) Let w0 be a nonextremal point of B, i.e. w0 = θu0+(1−θ)v0, with θ ∈ (0, 1),

u0, v0 ∈ B, u0 ≡ v0. Then, by continuity and the strict convexity of f , we haveθf(u(·, t))+(1−θ)f(v(·, t)) ≡ f(w(·, t)) for t ∈ [0, τ ], with τ > 0 small. By (19.23)and the strong maximum principle, we deduce that

w(x, τ) < w(x, τ) in Ω and∂w

∂ν(x, τ) >

∂w

∂ν(x, τ) on ∂Ω.

Due to (51.28), we know that for small τ > 0, the map L∞(Ω) u0 → u(·, τ ; u0) ∈C1(Ω) is well-defined and continuous on a neighborhood of u0. Therefore, if‖w0 − u0‖∞ is small enough, then Tmax(u0) > τ and u(τ) ≤ w(τ), where u :=u(·; u0). This, along with u(t) ≥ e−tAu0 guarantees that u0 ∈ B and w0 is aninterior point. The same argument applies for G.

To justify the last part of assertion (iii), write u0 = θv0 + (1− θ)v0, with v0 :=(1 − θ)−1(u0 − θv0) ≡ v0. For θ > 0 small, we have v0 ≥ v0 ≥ −(1− θ)−1θv0 ∈ Ddue to Theorem 19.2, hence v0 ∈ B (resp., G) by comparison.

(iv) Let u0 ∈ int(B). In particular, there exists v0 ∈ B, with u0 ≤ v0, u0 ≡ v0.Denote by u, v the solutions of (19.1) with initial data u0, v0, respectively. Due toExample 53.7, we know that uniformly bounded solutions are relatively compactin X := H1 ∩ C0(Ω) for t ≥ 1 and that the ω-limit set ω(u0) (in the X-topology)is nonempty and consists of equilibria. Let z ∈ ω(u0). By definition, there existsa sequence tk →∞, such that u(tk) → z in X . Since v(t) : t ≥ 1 is precompactin X , there exist z ∈ ω(v0) and a subsequence tkj such that v(tkj ) → z in X . ByLemma 17.9, there exist τ > 0 and α > 1, such that v ≥ αu for t ≥ τ > 0, hence

z ≥ αz. (19.26)

Due to f ≥ 0, we have z ≥ 0 by the maximum principle. Since z, z are steady statesof (19.1), we then deduce that z ≡ 0, because otherwise (19.26) would contradictProposition 19.8. Consequently, u0 ∈ D. In particular, int(B) ⊂ D. Conversely, itis clear that D ⊂ int(B) since D is open.

Remark 19.10. Instability of positive equilibria. Theorem 19.9 shows thatthe only way for the ω-limit set of a global bounded solution of (19.1) to containpositive equilibria is to have u(t) be an extremal (or boundary) point of B for allt ≥ 0. By the same token, if v is a positive steady state and u0 ≥ v, u0 ≡ v, thenu0 ∈ B. In the special case f(u) = |u|p, we have the following stronger property(which generalizes Theorem 17.8).

Proposition 19.11. Consider problem (19.1) with Ω bounded and f(u) = |u|p,p > 1. Assume that v0 ∈ B \ D and that u0 ≥ v0, u0 ≡ v0. Then Tmax(u0) <∞.

Proof. Let u, v be the corresponding solutions and assume for contradiction thatu is global. Then, by Lemma 17.9, we have u(t) ≥ αv(t), for all t ≥ 1 and someα > 1. On the other hand, by Example 53.7, ω(v0) contains a nonzero steady statez and there exists a sequence tk → ∞ such that v(tk) → z in C1(Ω). Moreover,since v(tk) ≥ e−tAv0, we have z ≥ 0 in Ω, hence ∂z/∂ν < 0 on ∂Ω by the Hopfmaximum principle. It follows that for large k, v(tk) > α−1z in Ω. Consequently,u(tk) > z, contradicting Theorem 17.8.

Remark 19.12. Further properties of D,B,G. Consider problem (19.1) withf(u) = |u|p and p > 1, and let us restrict ourselves to nonnegative initial data.Let us define G+ := u ∈ G : u ≥ 0 and B+,D+ similarly. We first note that theset D+ is unbounded, due to the existence of nonnegative global classical solutionssuch that ‖u(t)‖∞ → ∞ as t → 0+. Indeed, since 0 is an asymptotically stablesolution of problem (15.1) in Lq for q ≥ qc by Theorem 19.3, this occurs for any0 ≤ u0 ∈ Lq(Ω) \L∞(Ω) with ‖u0‖q small enough. On the other hand, for p < pS ,it follows from Theorem 6.2 that B+ = D+. Further related results will be obtainedlater, in particular in Subsection 28.4, where we study the transition between globalexistence and blow-up along each ray of nonnegative initial data starting from 0.Among the consequences of these results, let us mention the following properties:

(a) if p < pS, then G+ = B+ = D+ and G+ is a closed subset of L∞(Ω)(cf. Theorem 22.1);

(b) if p ≥ pS and Ω is starshaped, then B+ = D+ (cf. Corollary 5.2 andTheorem 28.7(iv));

(c) if p = pS and Ω is a ball, then G+ = B+ = D+ (cf. Theorem 28.7);

(d) if p > pS, Ω is a ball and we consider radial solutions only, then G+ =B+ = D+ (see Theorem 22.4).

Remark 19.13. Stabilization towards an equilibrium. In the proofs of The-orem 19.9 and Proposition 19.11 we used the fact that the ω-limit set of any globalbounded solution of (19.1) in a bounded domain Ω is a nonempty compact con-nected set consisting of equilibria. If all equilibria (at a given energy level) areisolated, then this fact guarantees that each global bounded solution converges toa single equilibrium. If n = 1 or if we consider radial solutions in a ball, then thisconvergence is true without any information on the set of equilibria, see [543], [353],[265], [102], [268], [127]. Similar stabilization result is true for general bounded do-mains in R

n, n ≥ 1, provided the nonlinearity is analytic, see [474], [291]. Othersufficient conditions can be found in the survey article [421]. Nonconvergent globalbounded solutions were constructed in [426] and [427] for spatially inhomogeneousnonlinearities of the form f = f(x, u).


Remark 19.14. Global solutions and very weak stationary solutions.Consider problem (14.1), where f : [0,∞)→ (0,∞) is a C1 nondecreasing convexfunction satisfying the blow-up condition (17.4). It was shown in [94] that strongrelations exist between the existence of global (classical) solutions of (14.1) andthe existence of very weak solutions of the stationary problem (13.1) (throughoutthis remark, “solution” implicitly means “nonnegative solution”):

(i) if Tmax(u0) =∞ for some 0 ≤ u0 ∈ L∞(Ω), then (13.1) admits a very weaksolution;

(ii) conversely, if (13.1) admits a very weak solution v, then for any u0 ∈ L∞(Ω)with 0 ≤ u0 ≤ v, we have Tmax(u0) = ∞.

Note that (i) provides a further blow-up criterion: if (13.1) has no very weaksolution, then all solutions of (14.1) have to blow up in finite time. As for (ii), itgives a new sufficient condition for global existence (see Theorem 20.5 for a relatedresult concerning the Cauchy problem).

Assertion (i) is not immediate since no bound is assumed on u. As for assertion(ii), it would be a direct consequence of the comparison principle if we were as-suming v ∈ L∞(Ω), but it is far from obvious in general since the inequality u ≤ vin itself does not a priori prevent ‖u(t)‖∞ from blowing up in finite time.

On the other hand, the existence of a global solution of (14.1) does not ingeneral imply the existence of a classical steady state. In fact, there are situationswhere (13.1) has a singular (very weak) solution but no classical solution (seeRemark 3.7) and where (14.1) admits global unbounded solutions which stabilizeto a singular solution as t→∞ (see Remark 22.6(b)).

The idea of the proof of assertion (i) is as follows. Assume u0 = 0 without lossof generality (u is then also global by the comparison principle). Since ut ≥ 0 byProposition 52.19, we may let v(x) := limt→∞ u(x, t) ≤ ∞. Theorem 17.3 implies∫Ω

u(t)ϕ1 dx ≤ C for t ≥ 0. Integrating (17.5) in time between t and t + 1 andusing ut ≥ 0, it follows that∫

Ω

f(u(t))ϕ1 dx ≤∫ t+1

t

∫Ω

f(u)ϕ1 dx ds

= λ1

∫ t+1

t

∫Ω

uϕ1 dx ds +[∫

Ω

u(s)ϕ1 dx]t+1

t≤ (1 + λ1)C.

Let now Θ ∈ C2(Ω), Θ ≥ 0, be the classical solution of the problem

−∆Θ = 1 in Ω,

Θ = 0 on ∂Ω.

(19.27)

Multiplying the equation in (14.1) by the function Θ defined in (19.27), integratingover Ω× (t, t + 1), and using ut ≥ 0, we obtain∫

Ω

u(t) dx ≤∫ t+1

t

∫Ω

u dx ds =∫ t+1

t

∫Ω

f(u)Θ dx ds−[∫

Ω

u(s)Θ dx]t+1

t≤ C′.

In particular, we get f(v) ∈ L1δ(Ω) and v ∈ L1(Ω). By arguing similarly as in the

(alternative) proof of Lemma 53.10, using again ut ≥ 0, we then easily concludethat v is a very weak solution of (13.1).

The proof of assertion (ii) is more delicate and will not be given here. It isbased on a perturbation argument which relies on a variant of Lemma 27.4 andon Lemma 27.5 below (used in the study of complete blow-up).

19.3. Diffusion eliminating blow-up

In Section 17, we used the convexity of the function f(u) = λu + up, u > 0, inorder to prove blow-up of solutions of (17.1) for suitable initial data. On the otherhand, it follows from Theorem 19.2 that any solution of (17.1) with Ω bounded,λ < λ1 and u0 small does exist globally and tends to zero as t → ∞. A similarassertion is true for Ω = R

n if, for example, λ = 0 and p > pF (see Theorem 18.1).Since all positive solutions of the ODE U ′ = Up blow up in finite time, we see thatdiffusion and the Dirichlet boundary conditions (or just the diffusion if Ω = R

n)can prevent blow-up for some initial data. Next we show that for some particularnonlinearities f , diffusion with the Dirichlet boundary condition can completelyeliminate blow-up. This result (and its modification for unbounded domains) isdue to [197].

Hence, let f : [0,∞) → [0,∞) be smooth, f(u) > 0 for u > 0 and consider theODE

Ut = f(U), t > 0,

U(0) = U0,(19.28)

where U0 > 0, and the related Cauchy-Dirichlet problem

ut − d∆u = f(u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (19.29)

where Ω ⊂ Rn is a bounded domain, d > 0 and u0 ≥ 0 is a an L∞-function. It

is well known that condition (17.4) is sufficient and necessary for blow-up of thesolution of (19.28), and we have seen in Theorem 17.3 that if f satisfies (17.4)and is convex, then the solution of (19.29) blows up for large initial data. We willprove that there exist (nonconvex) f satisfying (17.4) such that (19.29) possessesa global and bounded solution for any u0 and any d > 0.

Theorem 19.15. There is a C∞-function f : [0,∞) → [0,∞), f(u) > 0 foru > 0, such that the following holds:

(i) All solutions of (19.28) with U(0) > 0 blow up in finite time.

(ii) If Ω is bounded and d > 0, then all solutions of (19.29) with 0 ≤ u0 ∈ L∞(Ω)exist and remain bounded for all t ≥ 0.

Of course Theorem 19.15 cannot be true in the case of Neumann boundaryconditions uν = 0, since solutions of (19.28) also solve the PDE. On the otherhand, Theorem 19.15 remains true in the case of Robin boundary conditionsκuν + (1 − κ)u = 0, κ ∈ (0, 1), see [196].

The idea of the construction of the function f for Theorem 19.15 is to startwith a typical blow-up function satisfying (17.4), like f(u) = cup, with c > 0,p > 1, and then to modify it in an infinite number of intervals Ik = (ak, bk)with ak < bk < ak+1, ak → ∞. The modified function will be small enough insubintervals of Ik in order to provide us with suitable supersolutions of (19.29)but it will still satisfy condition (17.4).

Lemma 19.16. Let ak be an increasing sequence, a1 ≥ 1, limk→∞ ak = ∞.Then there are a C∞ function f : [0,∞) → [0,∞) with f(u) > 0 for u > 0, anda sequence bk such that

ak < bk < ak+1, (19.30)∫ ∞

1

du

f(u)<∞, (19.31)∫ bk

ak

du√F (bk)− F (u)

≥ k, (19.32)

for k = 1, 2, . . . , where F ′ = f .

Proof. Take any C1-function g : [0,∞) → [0,∞), with g(u) > 0 for u > 0, suchthat ∫ ∞

1

ds

g(s)<∞, g(s) ≥ 1 for s ≥ 1.

Choose also a positive sequence βk such that∑k

βk < ∞, βk < k2, 2β2kg(ak)k−2 < ak+1 − ak,

and define

γk := 1− βkk−2 > 0, bk := ak + β2kg(ak)k−2 < ak+1.

We will also choose sequences ck and dk (ak < bk < ck < ak+1, dk > 0) spec-ified later. Then, we construct an auxiliary function g by modifying the functiong on the intervals on [ak, bk] and [bk, ck] in the following way (see Figure 8)

g(u) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩dk +

g(ak)− dk

(bk − ak)γk(bk − u)γk for ak ≤ u ≤ bk,

dk +g(ck)− dk

ck − bk(u− bk) for bk ≤ u ≤ ck.


0 ak bk ck ak+1 bk+1 ck+1 u

g(u)

Figure 8: Graph of g.

Set also

G(u) = dk(u − bk)− g(ak)− dk

(γk + 1)(bk − ak)γk(bk − u)γk+1

on the interval [ak, bk]. Then(G)′ = g and∫ bk

ak

du√G(bk)− G(u)

=∫ bk

ak

(dk(bk − u) +

g(ak)− dk


)−1/2

du

→∫ bk

ak

(g(ak)


)−1/2

du

as dk → 0. Thus we obtain

limdk→0

∫ bk

ak

du√G(bk)− G(u)

=2

1− γk

√(γk + 1)(bk − ak)

g(ak).

We choose dk ∈ (0, 1/2) small enough so that∫ bk

ak

du√G(bk)− G(u)

≥ 11− γk

√bk − ak

g(ak)= k. (19.33)

Using g(s) ≥ 1 > 2dk > 0 for s ≥ 1 we obtain∫ bk

ak

du

g(u)=∫ bk

ak

(dk +

g(ak)− dk

(bk − ak)γk(bk − u)γk

)−1

du

≤∫ bk

ak

(g(ak)− dk

(bk − ak)γk(bk − u)γk

)−1

du

=bk − ak

(1 − γk)(g(ak)− dk)=

g(ak)βk

g(ak)− dk≤ 2βk,

and ∫ ck

bk

du

g(u)=∫ ck

bk

(dk +

g(ck)− dk

ck − bk(u − bk)

)−1

du

=ck − bk

g(ck)− dklog

g(ck)dk

≤ 2(ck − bk) logg(ck)dk

≤ βk

provided ck ∈ (bk, ak+1) is sufficiently close to bk. The above estimates imply∫ ck

ak

du

g(u)≤ 3βk.

This inequality and (19.33) guarantee that (19.31) and (19.32) are satisfied for g.Take a C∞-function f such that

12g(u) ≤ f(u) ≤ g(u). (19.34)

We can easily check (19.31). Since∫ bk

u

f(s)ds = F (bk)− F (u),

we haveF (bk)− F (u) ≤ G(bk)− G(u),

by integrating the second inequality in (19.34) over [bk, u]. This guarantees that falso satisfies (19.32).

The existence of supersolutions immediately follows from the previous lemma.

Lemma 19.17. Let f be as in Lemma 19.16 and d, L > 0. Then for sufficientlylarge k there is a solution uk of

d(uk)xx + f(uk) = 0 for − L < x < L, (19.35)

(uk)x(0) = 0, uk(x) ≥ ak for − L < x < L. (19.36)

20. Global existence for the Cauchy problem 129

Proof. Since the solution of the initial value problem

d(uk)xx + f(uk) = 0,

(uk)x(0) = 0, uk(0) = bk,

is given by ∫ bk

uk(x)

du√F (bk)− F (u)

=

√2d|x|,

the assertion follows from (19.32).

Proof of Theorem 19.15. Since Ω is bounded, we may choose L > 0 such that

x1 | x = (x1, x) ∈ Ω ⊂ [−L, L].

Let u0 ∈ L∞(Ω), u0 ≥ 0, and let u be the solution of (19.29). For large enoughpositive integer k, the function uk defined in Lemma 19.17 becomes a supersolutionof (19.29) and we have

u0(x) < ak ≤ uk(x1), x ∈ Ω.

Since there is no problem in comparing the data on the lateral boundary, thecomparison principle thus implies u(x, t) ≤ uk(x1) for t ∈ (0, Tmax(u0)), henceTmax(u0) =∞.

20. Global existence for the Cauchy problem

20.1. Small data global solutions

As announced in Section 18 (cf. Theorem 18.1(ii)), we show that, when p > pF ,small positive initial data yield global solutions. A simple example is providedby data dominated by a small multiple of a Gaussian, in which case the solutionremains controlled by the heat kernel. In all this section we use the notation (Gt)t>0

set in (48.5).

Theorem 20.1. Consider problem (18.1) with p > pF , u0 ∈ L∞(Rn), and letγ > 0. There exists ε = ε(γ) > 0 such that, if

0 ≤ u0(x) ≤ εGγ(x), x ∈ Rn, (20.1)

then Tmax(u0) = ∞ and u satisfies

u(x, t) ≤ CGt+γ(x), x ∈ Rn, t > 0 (20.2)

for some C = C(γ) > 0.

Theorem 20.1 is due to [220], where it was obtained by a contraction mappingargument. Here we shall derive Theorem 20.1 as a consequence of a more generalcriterion on u0 for global existence, due to [530].

Theorem 20.2. Consider problem (18.1) with p > pF . Assume that 0 ≤ u0 ∈L∞(Rn) satisfies ∫ ∞

0

‖e−sAu0‖p−1∞ ds < 1/(p− 1). (20.3)

Then Tmax(u0) = ∞ and u behaves like the solution of the linear part of theequation, up to multiplicative constants, i.e.:(

e−tAu0

)(x) ≤ u(x, t) ≤ C

(e−tAu0

)(x), x ∈ R

n, t > 0, (20.4)

for some C > 1 (depending on u0).

Remarks 20.3. (a) Inequality (20.2) corresponds in a sense to the minimalgrowth in time and space for positive solutions. Indeed, any positive solution u of(18.1) satisfies

u(x, t + τ) ≥ cGt+α(x), x ∈ Rn, t > 0

for some τ, α, c > 0 (this follows from the argument preceding formula (18.12)).(b) Since

‖e−tAu0‖∞ ≥ ct−n/2, t→∞ (20.5)

for any nontrivial u0 ≥ 0, it follows that condition (20.3) cannot be satisfied forp ≤ pF .

(c) A different smallness condition on u0 ensuring global existence appears inCorollary 20.20 below.

(d) The constant C in (20.4) can be explicitly computed from the proof below.In particular C converges to 1 as the LHS of (20.3) goes to 0.

Proof of Theorem 20.2. We look for a supersolution of the form

u(x, t) = h(t)(e−tAu0

)(x), x ∈ R

n, t > 0,

where

h(t) =(

1− (p− 1)∫ t

0

‖e−sAu0‖p−1∞ ds

)−1/(p−1)

.

Since

h′(t) = ‖e−tAu0‖p−1∞

(1− (p− 1)

∫ t

0

‖e−sAu0‖p−1∞ ds

)−1/(p−1)−1

= ‖e−tAu0‖p−1∞ hp(t),

it follows that

ut = h(t)(e−tAu0

)t+h′(t)e−tAu0 = ∆u + ‖e−tAu0‖p−1

∞ hp(t)e−tAu0 ≥ ∆u + up.

Since u(x, 0) = u0(x), we infer from the comparison principle that

0 ≤ u(x, t) ≤ u(x, t), x ∈ Rn, t < Tmax(u0).

The conclusion follows.

Proof of Theorem 20.1. By (48.6) we have e−tAGγ = Gt ∗ Gγ = Gt+γ . Since‖Gt+γ‖p−1

∞ = (4π(t + γ))−n(p−1)/2 and n(p − 1)/2 > 1, we deduce that (20.3) issatisfied with strict inequality for ε > 0 small. The conclusion then follows fromTheorem 20.2.

Remarks 20.4. (i) Global existence under assumption (20.1) can be shown by asimpler comparison argument, by looking for a supersolution of the form v(x, t) =ηtαG(x, t), where α, η > 0. Using ∂tG−∆G = 0, we obtain

vt −∆v − vp = ηαtα−1G− ηptαpGp

= ηtα−1[α− ηp−1t1+α(p−1)−n(p−1)/2e−(p−1)|x|2/(4t)

]G ≥ 0,

provided we choose α = (n/2)− 1/(p− 1) > 0 and η = α1/(p−1). It then suffices tocompare u with v(x, t + γ). However, this argument does not yield estimate (20.2)nor the sharp decay rate in t−n/2.

(ii) Let Ω ⊂ Rn be an arbitrary domain and e−tA denote the Dirichlet heat

semigroup in Ω. Then, for any p > 1 and u0 satisfying condition (20.3) (with0 ≤ u0 ∈ L∞(Ω), say), problem (15.1) has a unique global nonnegative (mild)solution and estimate (20.4) is true for x ∈ Ω and t > 0.

Indeed, the local in time solution u is constructed by the Banach fixed pointtheorem as a limit of iterations uk+1 = Φu0(uk), u1(t) ≡ 0 (cf. (15.12)). Butone easily shows that the function u(t) in the proof of Theorem 20.2 satisfiesu ≥ Φu0(u). By induction, it follows that u ≥ uk.

(iii) If p ≥ pS , then problem (18.1) possesses positive stationary solutions (seeTheorem 9.1). If pF < p and p(n − 4) < n, then the existence of global positivesolutions of (18.1) with exponentially decaying initial data also follows from Exam-ple 51.24 (the zero solution of (18.14) is exponentially stable), cf. Proposition 20.13and Remark 20.14(ii) below.

When n ≥ 3 and p > psg, we have a simple global existence criterion, forsolutions starting below the singular steady state (cf. [232, Theorem 10.4], wherea more general result is proved).

Theorem 20.5. Consider problem (18.1) with n ≥ 3, p > psg, and u0 ∈ L∞(Rn).Assume that |u0| ≤ u∗ in R

n \ 0, where u∗(x) := U∗(|x|) is defined in (3.9).Then Tmax(u0) = ∞.

Proof. The proof is based on the strong maximum principle, along with a space-shift argument. Assume for contradiction that T := Tmax(u0) < ∞.

We first claim that

u(x, t) ≤ u∗(x), x = 0, 0 < t < T. (20.6)

Fix 0 < τ < T . Since u is bounded in Rn×(0, τ), there exists ε > 0 such that u ≤ u∗

in 0 < |x| ≤ ε × (0, τ). By applying the comparison principle in the domain|x| > ε, it follows that u ≤ u∗ in (Rn \ 0)× (0, τ), hence (20.6). In particular,by parabolic estimates, u extends to a continuous function in (Rn \ 0)× (0, T ].

Now fix 0 < t0 < T . There exist a, ε > 0 such that

u(x, t0) ≤ u∗(x)− ε, 0 < |x| ≤ 3a. (20.7)

As a consequence of (20.6), (20.7) and of the strong maximum principle, appliedin the domain a < |x| < 3a, we deduce that

u(x, t) ≤ u∗(x) − η, |x| = 2a, t0 ≤ t ≤ T (20.8)

for some η > 0. By (20.7), (20.8) and continuity, one can find b ∈ Rn, 0 < |b| < a,

such that v(x, t) := u(x + b, t) satisfies

v(x, t0) < u∗(x), 0 < |x| ≤ 3a

andv(x, t) < u∗(x), |x| = 2a, t0 ≤ t ≤ T.

Since v is a solution, arguing as for (20.6), we deduce that

v(x, t) ≤ u∗(x), 0 < |x| ≤ 2a, t0 ≤ t < T,

henceu(x, t) ≤ u∗(x− b), 0 < |x| ≤ a, t0 ≤ t < T. (20.9)

Finally, (20.6) together with (20.9) imply supRn×(t0,T ) u < ∞. Applying the aboveargument to −u we obtain sup

Rn×(t0,T ) |u| < ∞ which contradicts T < ∞.

We have seen in Theorem 17.12 that if the nonnegative initial data decays slowerthan |x|−2/(p−1), then the solution of (18.1) blows up in finite time. We shallnow show that if the initial data decays faster (and satisfies a global smallnesscondition), then the solution exists for all times. Moreover, we shall show thatthese global solutions exhibit a typical parabolic feature: they have a temporaldecay whose exponent is precisely half that of the spatial decay of the initial data,with an upper limit of n/2. The following result is due to [324].

Theorem 20.6. Consider problem (18.1) with p > pF , and let k ≥ 2/(p − 1).There exists c = c(n, p, k) > 0 such that, if u0 ∈ L∞(Rn) satisfies

0 ≤ u0(x) ≤ c(1 + |x|)−k, x ∈ Rn, (20.10)

then Tmax(u0) = ∞ and we have, for all t ≥ 1,

‖u(t)‖∞ ≤

⎧⎪⎨⎪⎩t−n/2 if k > n,t−n/2 log t if k = n,t−k/2 if 2/(p− 1) ≤ k < n.

If moreover k > 2/(p− 1), then (20.4) is satisfied.

Remarks 20.7. (i) The decay rates in Theorem 20.6 are sharp for the choiceu0(x) = c(1 + |x|)−k. This follows from u(x, t) ≥ (e−tAu0)(x) and the lower esti-mates in Lemma 20.8.

(ii) For any p > 1, problem (18.1) admits some nontrivial global classical solu-tions. Of course, they have to change sign if p ≤ pF . For instance, for any p > 1,there exist self-similar solutions of the form u(x, t) = (t + 1)−1/(p−1)w(x/

√t + 1),

with w ∈ L∞(Rn) (see [269, Theorem 5]).

(iii) All the solutions constructed in Theorem 20.6 decay at least like t−1/(p−1).We shall see in Section 26 (see Theorem 26.9) that if p is less than a suitableexponent, then this is actually true for any nonnegative global classical solutionof (18.1), cf. also Theorem 28.10.

On the other hand, if p ≥ pJL and u∗(x) = cp|x|−2/(p−1) denotes the singularsteady state (see (3.9)), then there are bounded positive initial data u0 satisfyingu0 < u∗ such that the corresponding solutions exist globally, decay to zero, but

limt→∞ t1/(p−1)‖u(t)‖∞ =∞,

see [261]. In addition, if p > pJL and k ∈ (0, 2/(p − 1)) , then one can find > 2/(p−1) such that for any bounded nonnegative continuous radial u0 satisfyingu0 ≤ u∗ and u0(x) − u∗(x) ∼ |x|− for large |x|, there exist C1, C2 > 0 such that

C1t−k/2 ≤ ‖u(t)‖∞ ≤ C2t

−k/2, t ≥ 1, (20.11)

see Section 29. Recent results (see [306]) indicate that similar behavior of suitablepositive solutions can also be expected for p = pS provided n < 6 (if n = 4, thenthe decay rate t−k/2 should be replaced by t−k/2 log t).

(iv) If p < pS , u0 ≥ 0 has exponential decay (more precisely, u0(x)e|x|2/8 ∈

L2(Rn)) and the corresponding solution u exists globally, then (20.11) is true witheither k = n or k = 2/(p − 1) (and both possibilities occur). This follows fromTheorem 28.9 below.

For the proof of Theorem 20.6 we need the following lemma concerning thelinear heat equation. Here and in the rest of this subsection, f(t) ∼ g(t) meansthat

C1g(t) ≤ f(t) ≤ C2g(t) for some constants C1, C2 > 0.

Lemma 20.8. Let φ(x) = (1 + |x|)−k with k > 0. There holds

‖e−tAφ‖∞ ∼

⎧⎪⎨⎪⎩t−n/2 if k > n,t−n/2 log t if k = n,t−k/2 if k < n,

for t ≥ 1.

Proof. Due (1 + |x|)−k ≤ (1 + |x|2)−k/2 ≤ C(1 + |x|)−k, we may replace φ byφ(x) := (1+ |x|2)−k/2. For each t > 0, the function e−tAφ(x) is radially symmetricin x and nonincreasing in r = |x| (see Proposition 52.17). Consequently, we have

‖e−tAφ‖∞ = (e−tAφ)(0) =∫|y|≤1

(4πt)−n/2e−|y|2/4tφ(y) dy

+∫|y|>1

(4πt)−n/2e−|y|2/4tφ(y) dy =: I1(t) + I2(t).

First it is clear that I1(t) ∼ t−n/2 for t ≥ 1. If k > n, the conclusion follows fromI2(t) ≤ t−n/2

∫|y|>1

|y|−k dy = Ct−n/2. Now assume k ≤ n and observe that

I2(t) = π−n/2

∫|z|>1/(2

√t)

e−|z|2(1 + 4|z|2t)−k/2 dz

∼ t−k/2

∫|z|>1/(2

√t)

e−|z|2 |z|−k dz,

for t ≥ 1. If k < n, we simply use∫

Rn e−|z|2 |z|−k dz < ∞, hence I2(t) ∼ t−k/2

for t ≥ 1. If k = n, we use∫|z|>R

e−|z|2|z|−n dz =∫∞

Re−r2

r−1 dr ∼ log(1/R) forR ∈ (0, 1/2], hence I2(t) ∼ t−n/2 log t for t ≥ 1. The lemma follows.

Proof of Theorem 20.6. In view of the comparison principle, it is sufficient toprove the theorem when u0(x) = c(1 + |x|)−k for some c > 0.

Let us first consider the case k > 2/(p − 1). Since min(k, n)(p − 1)/2 > 1, itfollows from Lemma 20.8 that

∫∞0 ‖e−sAu0‖p−1

∞ ds < 1/(p−1) for c = c(n, p, k) > 0small enough. The result is then a consequence of Theorem 20.2.

Let us turn to the case k = 2/(p − 1). If n ≥ 3 and p > n/(n − 2), the resultfollows from the observation that the function u(x, t) = ε(1+ |x|2 +εt)−1/(p−1) is a(self-similar) supersolution for ε > 0 sufficiently small (which can be checked by asimple computation). If (n− 2)p ≤ n (or in the general case p > pF ), the result isa consequence of Theorem 20.19(i) and (ii) below (applied with u0 = c|x|−2/(p−1)

and v0 = c min(1, |x|−2/(p−1))) and of the comparison principle.

Similar results as in Theorem 20.6 can also be obtained for sign-changing solu-tions. Let us first prove two auxiliary lemmas concerning the linear heat equationon a half-line and in a cone in R

2.

Lemma 20.9. Let k ∈ [1, 2) and φ(x) = (1 − e−x)(1 + x)−k for x ≥ 0. Let e−tA

denote the Dirichlet heat semigroup in (0,∞). Then

‖e−tAφ‖∞ ∼ t−k/2, t ≥ 1.

Proof. We will use the formula

e−tAφ(x, t) =1√4πt

∫ ∞

0

(1− e−xy/t

)e−|x−y|2/4tφ(y) dy

and estimates

(1− e−xy/t)φ(y) ≤ min

(1,

xy

t

) 1(1 + y)k

, x, y, t > 0,

(1− e−xy/t)e−|x−y|2/4tφ(y) ≥ c

(1 + y)k≥ ct−k/2, for y ∈ [x, 2x], x2 = t ≥ 1,

where c > 0 denotes a generic constant which may change from step to step.If x2 = t ≥ 1, then the above estimates imply

e−tAφ(x, t) ≥ 1√4πt

∫ 2x

x

ct−k/2 dy = ct−k/2.

On the other hand, if t ≥ x2, t ≥ 1, then

e−tAφ(x, t) ≤ 1√4πt

(∫ x

0

xy

t

1(1 + y)k

dy +∫ ∞

x

e−|x−y|2/4t x

t

1yk−1

dy)

≤ c

t

∫ x

0

1(1 + y)k−1

dy + cx2−k

t

∫ ∞

x

1√te−|x−y|2/4t dy ≤ ct−k/2.

Finally, if 1 ≤ t < x2, then

e−tAφ(x, t) ≤ 1√4πt

(∫ x/2

0

e−x2/16t x

t

1(1 + y)k−1

dy

+∫ ∞

x/2

e−|x−y|2/4t c

(1 + x/2)kdy)

≤ c( x2

16t

)(3−k)/2

e−x2/16tt−k/2 +c

xk≤ ct−k/2,


In the following lemma we will use polar coordinates (r, ϕ) in R2.

Lemma 20.10. Let k ∈ N \ 0, Ω = Ωk := (r, ϕ) : r > 0, ϕ ∈ (−π/2k, π/2k).Then there exists u0 ∈ L∞(Ω), u0 ≥ 0, such that ‖e−tAu0‖∞ ∼ t−1−k/2 for t ≥ 1.

Sketch of proof (see [363] for details). Let G(r, ρ; ϕ, ψ; t) denote the Dirichletheat kernel in Ω and w(r, ϕ, t) := G(r, 1; ϕ, 0; t + t0), where t0 > 0 is fixed. Then

w(r, ϕ, t)

=1

4π(t + t0)

2k−1∑j=0

(−1)j exp[− (r cosϕ− cos jπ/k)2 + (r sin ϕ− sin jπ/k)2

4(t + t0)

].

Set s(r, t) := w(r, 0, t). Then one can show that ‖w(·, ·, t)‖∞ = supr>0 s(r, t) and

s(r, t) = C01

t + t0exp(− r2 + 1

4(t + t0)

)( r

t + t0

)k[1 +

r

t + t0R( r

t + t0

)],

where C0 is a positive constant and R is bounded on bounded sets. Obviously,

s(√

t + t0, t) ≥ ct−1−k/2 for t ≥ 1.

On the other hand, one can also show that supr s(r, t) is attained at some rM (t)which satisfies rM (t) ≤ C

√t + t0, hence s(rM (t), t) ≤ Ct−1−k/2.

Theorem 20.11. Let n ≥ 3, p > 1 and α > 1/(p − 1). Then there exists u0 ∈L∞(Rn) such that the solution u of (18.1) is global and ‖u(t)‖∞ ∼ t−α for t ≥ 1.

Proof. If α ≤ n/2, then p > pF and the assertion follows from Theorem 20.6.

Let n/2 < α < 2. Then n = 3. Set γ := α/3 and φ(x) :=∏3

i=1 ψ(xi), where

ψ(r) := sign(r)(1 − e−|r|)(1 + |r|)−2γ . (20.12)

Let −Am denote the Laplacian in Rm. Then e−tA3φ(x, t) =

∏3i=1 e−tA1ψ(xi, t),

hence ‖e−tA3φ‖∞ ∼ t−α for t ≥ 1 due to Lemma 20.9 and the oddness ofe−tA1ψ(·, t). Now choosing u0 = εφ, ε > 0 small, we obtain the result from Re-mark 20.4(ii) used with Ω = (0,∞)3.

Finally, let α ≥ 2. Fix k ∈ N such that γ := α− 1− k/2 ∈ [1/2, 1) and considerthe cone Ωk and the function w(t) := e−tAu0 from Lemma 20.10. Extend thefunction w to R

2 × [0,∞) by w(r, ϕ, t) = −w(r, π/k − ϕ, t) for ϕ ∈ (π/2k, 3π/2k)and w(r, ϕ + 2jπ/k, t) = w(r, ϕ, t), j = 1, 2, . . . , k − 1. Then w = w(x1, x2, t)is a solution of the heat equation in R

2 and ‖w(t)‖∞ ∼ t−1−k/2 for t ≥ 1. Setφ(x) = w(x1, x2, 0)ψ(x3), where ψ is defined by (20.12). Then, similarly as above,e−tAnφ(x, t) = w(x1, x2, t)e−tA1ψ(x3, t), hence ‖e−tA3φ‖∞ ∼ t−α for t ≥ 1. Nowchoosing u0 = εφ, ε > 0 small, we obtain the result from Remark 20.3(d) usedwith Ω = Ωk × (0,∞)× R

n−3.

Remark 20.12. (i) Solutions with exponential time decay. In addition tothe solutions with polynomial time decay in Theorem 20.11 one can also easilyconstruct sign-changing global solutions with exponential time decay. In fact, let1 0, A = π/2

√λ and let w be the positive solution of the problem

w′′ + wp = 0 in (−A, A), w(−A) = w(A) = 0. Choose α ∈ (0, 1) and set

u0(x) :=

αw(x − 4kA) if x ∈ [(4k − 1)A, (4k + 1)A),−αw(x − (4k + 2)A) if x ∈ [(4k + 1)A, (4k + 3)A),

where k ∈ Z. Then the solution u of (18.1) with n = 1 satisfies ‖u(t)‖∞ ∼ e−λt.This follows from Theorem 51.19 and Theorem 19.9(iv).

(ii) Decay of global solutions. Let 1 1.

If the solution u of (18.1) is global, then ‖u(t)‖∞ → 0 as t→∞ (see Theorem 26.9).This result is also true for all nonnegative data lying in the energy space E (see[478] and Example 51.27), but it fails for sign-changing radial solutions (considerthe choice α = 1 in (i)).

(iii) Notice that the radially symmetric initial data of the form c(1 + |x|)−k,k ≥ 2/(p − 1), appearing in Theorem 20.6 belong to the energy space E :=u ∈ Lp+1(Rn) : ∇u ∈ L2(Rn) if p < pS (see Example 51.27 for the well-posedness in this space).

20.2. Global solutions with exponential spatial decay

We have seen in Theorem 20.11 and Remark 20.12 that there is a wide range ofpossibilities for the decay of global solutions of the Cauchy problem (18.1). In thissubsection we show that the situation is much simpler if we restrict ourselves tothe initial data with exponential spatial decay. More precisely, we will considerinitial data in the space H1

g (see (18.16)) and exponents p ∈ (1, pS). We will usethe rescaled solutions v (see (18.13)) and operator L (see (18.15)). As above, letλL

k = (n + k − 1)/2 denote the eigenvalues of L. In addition, we denote λL0 :=

1/(p− 1).

Proposition 20.13. (i) Let 1 1 + 1/λLk0

. If u is a globalsolution of (18.1) with u0 ∈ H1

g \ 0 and t0 > 0, then there exist C1, C2 > 0 andk ∈ 0 ∪ k0, k0 + 1, k0 + 2, . . . such that

C1t−λL

k ≤ ‖u(t)‖∞ ≤ C2t−λL

k , t ≥ t0. (20.13)

Conversely, if k = 0 or k ≥ k0, then there exists u0 ∈ H1g such that the corre-

sponding solution of (18.1) is global and satisfies (20.13).(ii) Let pF n/2. Then there existsM = 0 such that

‖u(t)−M(t + 1)−n/2e−|x|2/4(t+1)‖∞ ≤ C(t + 1)−q, t ≥ 0.

Remark 20.14. (i) If p = 1 + 1/λLk0

for some k0 ≥ 1, then the proof of Propo-sition 20.13(i) guarantees the following: Let u be a global solution of (18.1) withu0 ∈ H1

g \ 0, t0 > 0. Then ‖u(t)‖∞ ≤ Ct−λL0 for t ≥ t0. If there exist C > 0

and λ > λL0 such that ‖u(t)‖∞ ≤ Ct−λ for t ≥ t0, then there exist C1, C2 > 0

and k > k0 such that (20.13) is true. Conversely, if k = 0 or k > k0, then thereexists u0 ∈ H1

g such that the corresponding solution of (18.1) is global and satisfies(20.13).

(ii) Some of the results in Proposition 20.13(i) concerning the decay (20.13) withk > 0 can also be obtained for supercritical p, p(n− 4) < n (cf. Example 51.24).

(iii) Sufficient conditions for the initial data u0 to satisfy the assumptions ofProposition 20.13(ii) can be found in Theorem 28.9. For related results see alsothe following subsection and [288], for example.

Proof of Proposition 20.13. (i) Let u be a global solution of (18.1) withu0 ∈ H1

g \ 0. Then the rescaled solution v (see (18.13)) is a global solutionof (18.14). If ‖v(s)‖H1

g→ 0 as s → ∞, then (20.13) is true with some k ≥ k0

due to Example 51.24 (see (51.72)). If ‖v(s)‖H1g → 0, then Lemma 18.4(iii) and

Example 51.24 show that ‖v(s)‖∞ ≤ C2 for all s ≥ s0 and s0 > 0.Assume lim infs→∞ ‖v(s)‖∞ = 0. Then the same estimates as at the end of

Example 51.24 guarantee lim infs→∞ ‖v(s)‖H1g

= 0. Consequently, choosing δ > 0small, there exist sj →∞ such that ‖v(sj)‖H1

g= δ. Using the compactness of the

semiflow we may assume v(sj) → w in H1g , where w belongs to the ω-limit set of

v, hence w is an equilibrium of (18.14), ‖w‖H1g

= δ. However, the zero equilibriumis isolated due to p /∈ 1+1/λL

k : k = 1, 2, . . . which yields a contradiction. HenceC1 ≤ ‖v(s)‖∞ ≤ C2 for s ≥ s0 which implies (20.13) with k = 0.

To prove the converse statement, assume first k = 0. By [174] there exist asequence of nontrivial stationary solutions vj , j = 1, 2, . . . , of problem (18.14).The corresponding rescaled solutions uj satisfy (20.13) with k = 0.

If k ≥ k0, then the existence of u0 ∈ H1g such that the solution u satisfies (20.13)

follows from Example 51.24.(ii) The proof is a direct consequence of assertion (ii) in Example 51.24.


20.3. Asymptotic profiles for small data solutions

More information on the asymptotic behavior of positive solutions than in Sub-section 20.1 can be gained if one considers suitably small initial data in L1.

Theorem 20.15. Consider problem (18.1) with p > pF . Assume that 0 ≤ u0 ∈L∞ ∩ L1(Rn) satisfies

‖u0‖1‖u0‖(n(p−1)/2)−1∞ ≤ c(n, p), (20.14)

with c(n, p) > 0 sufficiently small. Then Tmax(u0) = ∞ and (20.4) is satisfied.Moreover, u(t) behaves like a multiple of the heat kernel. Namely, the limit

I∞ := limt→∞ ‖u(t)‖1 exists and is finite, (20.15)

and there holds‖u(t)− I∞Gt‖1 → 0, t→∞. (20.16)

Theorem 20.15 is a variant of a result of [302] (see also [145], [322]). We prove(20.15), as a consequence of Theorem 20.2. This proof is simpler than those in [302](based on energy estimates) or in [145], [322] (based on the variation-of-constantsformula). As for property (20.16), it will be a consequence of the following lemmafrom [86] (see also [322]) concerning the inhomogeneous heat equation.

Lemma 20.16. Let u0 ∈ L1(Rn), f ∈ L1(Rn × (0,∞)) with u0, f ≥ 0, u0 ≡ 0and let u be given by

u(t) = e−tAu0 +∫ t

0

e−(t−s)Af(s) ds, t > 0.

Then M := limt→∞ ‖u(t)‖1 exists in (0,∞) and we have

‖u(t)−M Gt‖1 → 0, t→∞.

Proof. By the variation-of-constants formula, we have

‖u(t)− e−(t−t0)Au(t0)‖1 =∫ t

t0

‖f(s)‖1 ds, t ≥ t0 ≥ 0.

Since ‖e−(t−t0)Au(t0)‖1 = ‖u(t0)‖1 we see that limt→∞ ‖u(t)‖1 exists and is finite.Since u(t) ≥ e−tAu0 this limit is positive. Denoting M(t) = ‖u(t)‖1, it follows that

‖u(t)−M Gt‖1 ≤∫ ∞

t0

‖f(s)‖1 ds +∥∥e−(t−t0)Au(t0)−M(t0)Gt

∥∥1+|M(t0)−M |.


Using ∥∥e−sAφ− ‖φ‖1 Gs

∥∥1→ 0, s →∞, 0 ≤ φ ∈ L1(Rn)

(see Proposition 48.6 in Appendix B) and letting t→∞, we obtain

lim supt→∞

‖u(t)−M Gt‖1 ≤∫ ∞

t0

‖f(s)‖1 ds + |M(t0)−M |.

The lemma follows by letting t0 →∞.

Proof of Theorem 20.15. Assume (20.14) with c = c(n, p) > 0 small. Using theLp-Lq-estimate (cf. Proposition 48.4(d)) and choosing τ = (‖u0‖1/‖u0‖∞)2/n, weobtain ∫ ∞

0

‖e−sAu0‖p−1∞ ds ≤

∫ τ

0

‖u0‖p−1∞ ds +

∫ ∞

τ

‖u0‖p−11 s−n(p−1)/2 ds

= τ‖u0‖p−1∞ + C(n, p)‖u0‖p−1

1 τ1−n(p−1)/2

≤ C(n, p)‖u0‖2/n1 ‖u0‖p−1−2/n

∞ ≤ 1/2(p− 1).

By Theorem 20.2, we deduce that Tmax(u0) = ∞ and that

u(t) ≤ Ce−tAu0. (20.17)

Applying the Lp-Lq-estimate again, we obtain

‖u(t)‖pp ≤ C‖e−tAu0‖p

p ≤ C min(1, t−n(p−1)/2). (20.18)

On the other hand, by Proposition 48.4(b) and the variation-of-constants formula,we have ∫

Rn

u(t) dx =∫

Rn

u0 dx +∫ t

0

‖u(s)‖pp ds. (20.19)

We deduce from (20.18), (20.19) and n(p− 1)/2 > 1 that ‖u(t)‖1 is nondecreasingand bounded. The conclusion then follows from Lemma 20.16.

Remark 20.17. Estimates similar to (20.16) are also true for other Lq-norms. Inparticular there holds

tn/2‖u(t)− I∞Gt‖∞ → 0, t→∞. (20.20)

This is a consequence of [70, Theorem 4.1] and inequality (20.17). Estimates forall Lq-norms follow immediately by interpolating between (20.16) and (20.20).

It follows from Theorems 17.12 and 20.6 that initial data which decay at therate |x|−2/(p−1) constitute a borderline between blow-up and global existence. Our


next results concern some particular classes of initial data with this asymptoticbehavior which are especially interesting.

First it turns out that initial data which are homogeneous of degree −2/(p− 1)(and suitably small) give rise to (forward) self-similar solutions, cf. Remark 15.4(ii).Moreover, these solutions enjoy stability properties. For instance, if a (small) initialdata coincides for large x with a homogeneous function of degree −2/(p − 1),then the solution is asymptotically self-similar. These results will be proved bysemigroup techniques and suitable fixed point arguments, that are refinements ofthe methods introduced in Section 15.

We will also describe the global properties of the equation in a space whichnaturally arises in this connection, namely the critical Lq-space. Its special role asan invariant space can be explained as follows (cf. e.g. [112], and see also [115] forearlier references and for similar considerations concerning the Navier-Stokes andnonlinear Schrodinger equations). Consider the scaling transformations

Sλ : u → uλ(x, t) := λ2/(p−1)u(λx, λ2t)

for λ > 0. Observe that the equation in (18.1) is invariant under these transfor-mations. On the other hand, for spatial functions φ = φ(x), we have

‖φλ‖q = λ2/(p−1)−(n/q)‖φ‖q, 1 ≤ q ≤ ∞, (20.21)

so that the only Lq-norm left invariant by the transformations Sλ is the criticalnorm, i.e. q = qc = n(p− 1)/2. Now assume that there exists q with the propertythat the solution of (18.1) is global whenever the initial data u0 is small in Lq.If q = qc, then, by (20.21) applied to φ = u0, global existence will hold for anyu0 ∈ Lq. But this is a contradiction to Theorem 17.1; hence q = qc is the onlypossible value with that property.

In accordance with these observations, we will indeed prove global existencefor small initial data in Lqc , provided that qc > 1. Note that the criticalexponent p = pF corresponds to the case when qc = 1, and the requirement thatqc > 1 is thus consistent with the Fujita-type result Theorem 18.1. Furthermore,still using the techniques mentioned in the previous paragraph, we will establishthe asymptotic stability of the zero solution in the space Lqc . More generally, wewill show that the above mentioned self-similar solutions are in a sense stable withrespect to critical Lq-perturbations. Note, in turn, that the transformations Sλ

also leave invariant the homogeneous functions of degree −2/(p− 1) (from whichthe self-similar solutions arise).

We shall use the following definition of mild solution of problem (18.1).

Definition 20.18. Let u0 ∈ L1(Rn) + L∞(Rn). We say that u is (global) mildsolution of (18.1) if u ∈ L∞

loc((0,∞), Lr(Rn)) for some r ≥ p and satisfies

u(t) = e−tAu0 +∫ t

0

e−(t−s)A|u|p−1u(s) ds, t > 0,

where for each t > 0 the integral is absolutely convergent in Lr(Rn). In particular,there holds u(t)− e−tAu0 → 0 in Lr(Rn) as t→ 0.

This definition is slightly different from that in Remark 15.4(iii). Note that thedefinition makes sense since e−(t−s)A|u|p−1u(s) ∈ L∞

loc((0, t), Lr(Rn)), due to r ≥ pand the Lp-Lq-estimates. The following result is due to [457], [115] for assertion(i), [115], [475] for assertion (ii). Assertion (iii) for u0 = 0 (i.e. Corollary 20.20) isfrom [481], improving on earlier results of [530], whereas the case u0 = 0 seemsnew.

Theorem 20.19. Let p > pF , ω ∈ L∞(Sn−1) and set

u0(x) := |x|−2/(p−1)ω(x/|x|), x ∈ Rn \ 0. (20.22)

There exists µ0 = µ0(n, p) > 0 such that, if ‖ω‖∞ ≤ µ0, then the following prop-erties hold.

(i) Problem (18.1) admits a global mild solution u (in the sense of Definition 20.18).Moreover, u is self-similar, i.e. is of the form

u(x, t) = t−1/(p−1)f(x/√

t), x ∈ R

n, t > 0,

with f(y) = u(y, 1) ∈ L∞(Rn), and u is a classical solution for t > 0. Furthermore,the solution u is stable in the sense indicated in parts (ii) and (iii) hereafter.

(ii) Let v0 = ϕu0, where ϕ ∈ L∞(Rn) satisfies ϕ = 1 for |x| large. Assume that‖ω(·/| · |)ϕ‖∞ ≤ µ0. Then problem (18.1) with initial data v0 admits a globalsolution v with v(t) ∈ L∞(Rn) for each t > 0, and v is a classical solution fort > 0. Furthermore, v is asymptotically self-similar, with profile f , i.e.:

t1/(p−1)‖u(t)− v(t)‖∞ = supy∈Rn

∣∣t1/(p−1)v(y√

t, t)−f(y)

∣∣→ 0, t→∞. (20.23)

(iii) Let q := qc. Assume that v0 ∈ L1(Rn) + L∞(Rn) satisfies u0 − v0 ∈ Lq(Rn)and

‖u0 − v0‖q < µ0.

Then problem (18.1) with initial data v0 admits a global solution v which satisfies(20.23), together with

supt>0

‖u(t)− v(t)‖q ≤ 2‖u0 − v0‖q (20.24)

and‖u(t)− v(t)‖q → 0, t→∞. (20.25)

Corollary 20.20. Let p > pF and q := qc. Then the zero solution is asymptoti-cally stable in Lq. More precisely, if v0 ∈ Lq(Rn) satisfies

‖v0‖q ≤ µ0

with µ0 = µ0(n, p) > 0 sufficiently small, then (18.1) admits a global mild solutionv which satisfies

supt>0

‖v(t)‖q ≤ 2‖v0‖q and ‖v(t)‖q → 0, t→∞.

Furthermore v(t) ∈ L∞(Rn) for each t > 0, v is a classical solution for t > 0, andthere holds

t1/(p−1)‖v(t)‖∞ → 0, t→∞.

Remarks 20.21. (i) Nonuniqueness. The solutions u and v constructed inTheorem 20.19 are unique in a suitable class of functions (see Lemma 20.22 andcf. also Remark 20.24(iii) below). When ω is a suitably small positive constant,nonuniqueness in a larger class of functions has been proved in [499]. Anothernonuniqueness result can be found in [389].

(ii) Decay rates. The convergence statement in Theorem 20.19(ii) says thatu(t)− v(t) decays in L∞ faster that u(t) or v(t) separately. More precise estimateson the decay of u(t)−v(t) when u is radial can be found in [205]. Observe that theasymptotic behaviors in Theorem 20.19 and in Corollary 20.20 are different (notethat u0 in Theorem 20.19 just fails to be in Lq for q = qc if u0 ≡ 0). In particular,in Theorem 20.19 with u0 ≡ 0, ‖v(t)‖∞ decays like t−1/(p−1) as t → ∞, whereasin Corollary 20.20 it decays faster.

(iii) Radial self-similar solutions. The self-similar solutions constructed inTheorem 20.19 are not radial unless u0 is radial. Radial self-similar solutions of(18.1) have been constructed by ODE or variational techniques (see Remarks 15.4and the references there). In the radial case, the decay of the profile f(y) as y →∞has also been studied. The profile can decay either like |y|−2/(p−1) or exponentially.

(iv) Other domains. Consider problem (15.1) in a (possibly unbounded) do-main Ω. By the comparison principle and Corollary 20.20, it follows that the zerosolution is asymptotically stable in Lq for q = qc. This is in contrast with thesituation for q > qc (cf. Theorem 19.3).

In view of the proof, we introduce the following notation. For p, q as above, wefix r such that

1 ≤ r/p < q < r. (20.26)

Although r is not uniquely determined, we assume that it is fixed once and for all(see also Remark 20.24(ii) below). We let β = n

2 (1q −

1r ) = 1

p−1 −n2r and we define

the following function spaces:

X =u ∈ L∞

loc((0,∞), Lr(Rn)) : ‖u‖X < ∞, where ‖u‖X = sup

t>0tβ‖u(t)‖r,

Y =u ∈ L∞

loc((0,∞), L∞(Rn)) : ‖u‖Y < ∞, where ‖u‖Y = sup

t>0t

1p−1 ‖u(t)‖∞,

and Z = X ∩ Y , with norm

‖u‖Z = ‖u‖X + ‖u‖Y .

For δ ≥ 0, we also define

Eδ =u0 ∈ L1(Rn) + L∞(Rn) : Nδ(u0) <∞

,

where Nδ(u0) = supt>0

tβ+δ‖e−tAu0‖r,(20.27)

and for 0 < T < ∞ we set

‖u‖X,δ,T = sup0<t<T

tβ+δ‖u(t)‖r < ∞, u ∈ X,

‖u‖Y,δ,T = sup0<t<T

t1

p−1+δ‖u(t)‖∞ <∞, u ∈ Y,

and‖u‖Z,δ,T = ‖u‖X,δ,T + ‖u‖Y,δ,T , u ∈ Z.

We note right away that for all 1 ≤ m ≤ q, due to the Lp-Lq-estimates (seeProposition 48.4), we have Lm(Rn) ⊂ Eδ and

Nδ(u0) ≤ ‖u0‖m, with δ = δ(m) =n

2m− 1

p− 1. (20.28)

Moreover, we set E := E0, N := N0 and for M > 0, we denote by BX(M)(resp., BY (M), BZ(M)) the closed ball of radius M in X (resp., Y, Z). The mainingredient of the proof of Theorem 20.19 is the following lemma.

Lemma 20.22. (i) There exists ε0 = ε0(n, p, r) > 0 such that if u0 ∈ E satisfiesN (u0) ≤ ε0, then (18.1) admits a unique global mild solution u ∈ BZ(M) withM = C(p)N (u0). Moreover u is a classical solution of (18.1) for t > 0.(ii) Let 0 ≤ δ < δ := np/2r − 1/(p − 1). There exists ε1 = ε1(n, p, r, δ) ∈ (0, ε0]such that if u0, v0 ∈ E satisfy N (u0),N (v0) ≤ ε1 and u0 − v0 ∈ Eδ, then thecorresponding solutions u, v of (18.1) given by part (i) satisfy

supt>0

[t

1p−1+δ‖u(t)− v(t)‖∞ + tβ+δ‖u(t)− v(t)‖r

]≤ C(p)Nδ(u0 − v0). (20.29)

(iii) Let m ∈ (r/p, q] and set δ = n/2m−1/(p−1). There exists ε2 = ε2(n, p, r, m) ∈(0, ε0] such that if u0, v0 ∈ E satisfy N (u0),N (v0) ≤ ε2 and u0 − v0 ∈ Lm(Rn),then the corresponding solutions u, v of (18.1) given by part (i) satisfy

supt>0

tδ‖u(t)− v(t)‖q ≤ 2‖u0 − v0‖m. (20.30)

Proof. For u0 ∈ L1(Rn) + L∞(Rn) and u ∈ X , we define the mapping

Tu0u(t) = e−tAu0 +∫ t

0

e−(t−s)A|u|p−1u(s) ds.

Let M > 0. We fix 0 ≤ δ < δ (≤ 1) and u0, v0 ∈ E, with u0 − v0 ∈ Eδ.Step 1. Estimates of the mapping T in X. For all u, v ∈ X and 0 < s < t <

T < ∞, we have∥∥e−(t−s)A(|u|p−1u(s)− |v|p−1v(s)

)∥∥r

≤ (t− s)−n(p−1)/2r∥∥ |u|p−1u(s)− |v|p−1v(s)

∥∥r/p

≤ (t− s)−q/r(‖u(s)‖p−1

r + ‖v(s)‖p−1r

)‖u(s)− v(s)‖r

≤ (t− s)−q/rs−(βp+δ)(‖u‖p−1

X + ‖v‖p−1X

)‖u− v‖X,δ,T

On the other hand, using 1− β(p− 1)− q/r = 0, we have

tβ+δ

∫ t

0

(t− s)−q/rs−(βp+δ) ds = t1−β(p−1)−q/r

∫ 1

0

(1− σ)−q/rσ−(βp+δ) dσ

= C(n, p, r, δ),

where the integrals are finite, due to q/r < 1 and βp + δ < βp + δ = 1. It followsthat

tβ+δ‖Tu0u(t)− Tv0v(t)‖r ≤ tβ+δ‖e−tA(u0 − v0)‖r

+ tβ+δ

∫ t

0

∥∥e−(t−s)A(|u|p−1u(s)− |v|p−1v(s)

)∥∥rds

≤ Nδ(u0 − v0) + C(‖u‖p−1

X + ‖v‖p−1X

)‖u− v‖X,δ,T ,

hence

‖Tu0u− Tv0v‖X,δ,T ≤ Nδ(u0 − v0) + C1Mp−1‖u− v‖X,δ,T , u, v ∈ BX(M),

(20.31)with C1 = C1(n, p, r, δ) > 0.

Step 2. Estimates of the mapping T in Z and fixed-point. For all u, v ∈ Z,0 < t < T < ∞ and t/2 < s < t, we have∥∥e−(t−s)A

(|u|p−1u(s)− |v|p−1v(s)

)∥∥∞

≤∥∥|u|p−1u(s)− |v|p−1v(s)

∥∥∞

≤ p[‖u(s)‖p−1

∞ + ‖v(s)‖p−1∞]‖u(s)− v(s)‖∞

≤ p(t/2)−p

p−1−δ(‖u‖p−1

Y + ‖v‖p−1Y

)‖u− v‖Y,δ,T .


Using the fact that

Tu0u(t) = e−(t/2)A(Tu0u(t/2)

)+∫ t

t/2

e−(t−s)A|u|p−1u(s) ds,

it follows that, for all u, v ∈ BZ(M),

t1

p−1+δ‖Tu0u(t)− Tv0v(t)‖∞≤ t

1p−1+δ

∥∥e−(t/2)A(Tu0u( t

2 )− Tu0v( t2 ))∥∥

∞

+ t1

p−1+δ

∫ t

t/2

∥∥e−(t−s)A(|u|p−1u(s)− |v|p−1v(s)

)∥∥∞ ds

≤ 21

p−1+δ(4π)−n2r ( t

2 )1

p−1− n2r +δ‖Tu0u( t

2 )− Tu0v( t2 )‖r

+ p21

p−1+δ(‖u‖p−1

Y + ‖v‖p−1Y

)‖u− v‖Y,δ,T

≤ C(p)‖Tu0u− Tu0v‖X,δ,T + C(p)Mp−1‖u− v‖Y,δ,T .

Taking supremum for t ∈ (0, T ) and combining this with (20.31), we obtain

‖Tu0u− Tu0v‖Z,δ,T ≤ C2Nδ(u0 − v0) + C3Mp−1‖u− v‖Z,δ,T , u, v ∈ BZ(M),

(20.32)with C2 = C2(p) ≥ 1 and C3 = C3(n, p, r, δ) > 0. In particular, letting T →∞ in(20.32) with δ = 0, we get

‖Tu0u− Tv0v‖Z ≤ C2N (u0 − v0) + C3Mp−1‖u− v‖Z , u, v ∈ BZ(M). (20.33)

Choose ε0 = ε0(n, p, r) > 0 such that 2pC3(n, p, r, 0)(C2ε0)p−1 ≤ 1 and assumethat N (u0) ≤ ε0. Taking M = 2C2N (u0), we have C3M

p−1 ≤ 1/2 and C2N (u0)+C3M

p ≤ M . It follows from (20.33) (with the choices v0 = 0, v = 0 and u0 = v0)that Tu0 is a strict contraction on the complete metric space BZ(M), endowedwith the distance induced by the norm ‖ ·‖Z . Therefore it possesses a unique fixedpoint, that we denote by u(t) = Wtu0. In particular u(t) ∈ L∞(Rn) for t > 0 andu is a classical solution of (18.1) for t > 0. This proves the existence-uniquenessstatement of assertion (i).

Next, assume in addition that N (v0) ≤ ε0 and put v(t) = Wtv0. Replacing ε0 byε1 > 0 possibly smaller and depending also on δ, we have C3(n, p, r, δ)Mp−1 ≤ 1/2.It then follows from (20.32) that

‖u− v‖Z,δ,T ≤ 2C2Nδ(u0 − v0).

Assertion (ii) follows by letting T →∞.Step 3. Lq-estimates. Fix m ∈ (r/p, q] and put δ = δ(m) = n

2m − 1p−1 . Note

that δ ∈ [0, δ). Assume that u0, v0 ∈ E satisfy N (u0),N (v0) ≤ ε2(n, p, r, m) :=

ε1(n, p, r, δ) and u0 − v0 ∈ Lm(Rn). Let u, v be the corresponding solutions of(18.1) given by Steps 1 and 2. Similarly as in the beginning of Step 1, we obtainfor 0 < s < t:∥∥e−(t−s)A

(|u|p−1u(s)− |v|p−1v(s)

)∥∥q

≤ (t− s)−n2 ( p

r − 1q )s−(βp+δ)

(‖u‖p−1

X + ‖v‖p−1X

)supσ>0

σβ+δ‖u(σ)− v(σ)‖r .

(20.34)On the other hand, using 1− n

2 (pr −

1q )− βp = 0, we have∫ t

0

(t− s)−n2 ( p

r − 1q )s−(βp+δ) ds = t−δ

∫ 1

0

(1 − σ)−n2 ( p

r − 1q )σ−(βp+δ) dσ

= C(n, p, r, δ)t−δ,

(20.35)

where the integrals are finite, due to n2 (p

r −1q ) < n(p − 1)/2q = 1 and βp + δ <

βp + δ = 1. Combining (20.34), (20.35) and (20.28) (and taking ε2(n, p, r, m)possibly smaller), we obtain

tδ‖u(t)−v(t)‖q

≤ tδ‖e−tA(u0 − v0)‖q + tδ∫ t

0

∥∥e−(t−s)A(|u|p−1u(s)− |v|p−1v(s)

)∥∥qds

≤ ‖u0 − v0‖m + C(n, p, r, δ)(‖u‖p−1

X + ‖v‖p−1X

)supσ>0

σβ+δ‖u(σ)− v(σ)‖r

≤ ‖u0 − v0‖m + C(n, p, r, δ)Mp−1Nδ(u0 − v0) ≤ 2‖u0 − v0‖m.

This proves assertion (iii).

The next lemma shows that the homogeneous initial data u0 belong to the classE used in Lemma 20.22.

Lemma 20.23. Let 0 < k < n, L > 0, and let u0 ∈ L1(Rn) + L∞(Rn) satisfy

|u0(x)| ≤ L|x|−k.

Then, for n/k < s ≤ ∞, there holds

supt>0

tk/2−n/(2s)‖e−tAu0‖s ≤ cL

where c = c(n, k, s) = ‖e−A|x|−k‖s < ∞.

Proof. Set φ(x) = |x|−k and decompose φ = φ1 + φ2, where φ1 = χ|x|<1φ,φ2 = χ|x|≥1φ. Then φ1 ∈ Lm(Rn), m < n/k and φ2 ∈ Ls(Rn), s > n/k.Consequently,

e−Aφ = e−Aφ1 + e−Aφ2 ∈ Ls(Rn), s > n/k.


Now using φ(λx) = λ−kφ(x), we obtain

|(e−tAu0)(x)| =∣∣∣∫

Rn

(4πt)−n/2e−|y|2/4tu0(x− y) dy∣∣∣

≤ (4π)−n/2L

∫Rn

e−|z|2/4φ(x − zt1/2) dz

= Lt−k/2(4π)−n/2

∫Rn

e−|z|2/4φ(xt−1/2 − z) dz

= Lt−k/2(e−Aφ

)(xt−1/2).

In particular,

‖e−tAu0‖s ≤ Lt(n/2s)−(k/2)‖e−Aφ‖s, s > n/k,

and the lemma follows.

Proof of Theorem 20.19. In this proof we shall take µ0 as small as necessaryto apply Lemma 20.22, but µ0 will depend only on n, p, r.

(i) Since N (u0) ≤ c(n, p, r)‖ω‖∞ by Lemma 20.23, the existence of u followsfrom Lemma 20.22(i).

Let us show that u is self-similar. This is equivalent to showing that, for eachλ > 0, uλ(x, t) := λ2/(p−1)u(λx, λ2t) satisfies uλ ≡ u (indeed, consider λ =t−1/2). Since ‖uλ‖X = ‖u‖X , it is thus sufficient, in view of the uniqueness partof Lemma 20.22(i), to check that uλ is also a mild solution of (18.1). To thisend, we define the dilation operator (dλf)(x) := f(λx) and note that uλ(t) =λ2/(p−1)dλu(λ2t). A direct computation involving the heat kernel yields

e−tA(dλf) = dλ

(eλ2−tAf

). (20.36)

Applying (20.36) with f = up, we see that the function

(Su)(t) :=∫ t

0

e−(t−s)Aup(s) ds

satisfies

S(uλ)(t) = λ2p/(p−1)

∫ t

0

e−(t−s)Adλup(λ2s) ds

= λ2p/(p−1)dλ

∫ t

0

eλ2−(t−s)Aup(λ2s) ds

= λ2p/(p−1)dλ

∫ λ2t

0

e−(λ2t−σ)Aup(σ)λ−2dσ

= λ2/(p−1)dλ(Su)(λ2t) =: (Su)λ(t).

(20.37)

Now, since u0 satisfies (20.22), we have dλu0 = λ−2/(p−1)u0, hence

(e−tAu0)λ := λ2/(p−1)dλ(eλ2−tAu0) = λ2/(p−1)e−tA(dλu0) = e−tAu0. (20.38)

Combining (20.37) and (20.38), it follows that

e−tAu0 + S(uλ)(t) = (e−tAu0)λ + (Su)λ(t) = uλ(t).

We have thus shown that u is self-similar.(ii) Since N (v0) ≤ c(n, p, r)‖ω(·/| · |)ϕ‖∞ by Lemma 20.23, the existence of v

follows from Lemma 20.22(i). Next, since |v0−u0| ≤ C|x|−2/(p−1)χ|x|<R for someC, R ∈ (0,∞) and 2/(p − 1) < n, we have v0 − u0 ∈ Lm(Rn) for all m ∈ [1, q),hence v0 − u0 ∈ Eδ for all δ ∈ (0, δ), by (20.28). The assertion then follows fromLemma 20.22(ii).

(iii) Using (20.28), we get

N (v0) ≤ N (v0 − u0) +N (u0) ≤ ‖v0 − u0‖q + c(n, p, r)‖ω‖∞ ≤ (1 + c(n, p, r))µ0.

The existence of a global solution v with initial data v0 is a consequence ofLemma 20.22(i). Property (20.24) is just (20.30) with m = q (hence δ = 0).

Let us show (20.25) and (20.23). To this end, we fix m ∈ (r/p, q), we let η :=u0−v0 and we introduce the sequence ηi := ηχ|x|<i ∈ Lm∩Lq(Rn), which satisfiesηi → η in Lq(Rn). Let ψi := u0 + ηi. We have ‖ψi − u0‖q = ‖ηi‖q ≤ ‖v0 − u0‖q.Consequently, the existence of a global solution vi of (18.1) with initial data ψi isensured. By (20.30) and (20.28) we have

‖vi(t)− u(t)‖q ≤ 2‖ηi‖mt−δ(m) and ‖vi(t)− v(t)‖q ≤ 2‖ηi − η‖q.

For each i, it follows that

lim supt→∞

‖u(t)−v(t)‖q ≤ lim supt→∞

‖u(t)−vi(t)‖q +lim supt→∞

‖vi(t)−v(t)‖q ≤ 2‖ηi−η‖q,

and property (20.25) follows by letting i → ∞. On the other hand, (20.29) and(20.28) imply

t1/(p−1)‖vi(t)− u(t)‖∞ ≤ C(p)Nδ(m)(ηi)t−δ(m) ≤ C(p)‖ηi‖mt−δ(m)

andt1/(p−1)‖vi(t)− v(t)‖∞ ≤ C(p)N (ηi − η) ≤ C(p)‖ηi − η‖q.

For each i, it follows that

lim supt→∞

t1/(p−1)‖u(t)− v(t)‖∞

≤ lim supt→∞

t1/(p−1)‖u(t)− vi(t)‖∞ + lim supt→∞

t1/(p−1)‖vi(t)− v(t)‖∞

≤ C(p)‖ηi − η‖q,

and property (20.23) follows by letting i →∞.

Remarks 20.24. (i) When u0 ∈ Lqc is not small, (18.1) still admits a local intime solution (cf. Remark 15.4(i)). The existence can be proved by argumentssimilar to those in the proof of Lemma 20.22(i).

(ii) It can be shown that the solution u constructed in Lemma 20.22 does notdepend on the choice of r if u0 is suitably small. More precisely, given another rsatisfying (20.26) and denoting by N = N0 the corresponding norm in (20.27),there exists ε0 ∈ (0, min(ε0(n, p, q, r), ε0(n, p, q, r))] such that the two solutionscoincide if N (u0),N (u0) ≤ ε0.

(iii) It follows from the proof of Lemma 20.22 that (18.1) admits a mild solutionwhich is unique in the larger class BX(K) with K = C(p)N (u0).

21. Parabolic Liouville-type results

In Section 18 on Fujita-type results, we have seen that the equation ut−∆u = up

with p > 1 has no global positive (classical) solution in Rn × (0,∞) if (and only

if) p ≤ pF . In view of the Liouville-type results proved in Section 8 for the ellipticequation−∆u = up, it is natural to look also for parabolic Liouville-type theorems.More precisely, if one now considers positive solutions that are global for bothpositive and negative time, i.e. solutions on the whole space R

n+1 = Rn × R, can

one prove nonexistence for a larger range of p’s than in the Fujita problem ? Wewill also study the same question on a half-space.

As it will turn out, we shall see in Section 26 that such results have manyapplications in the study of a priori estimates and (blow-up) singularities.

Let us first consider the case of radial solutions, for which we have the followingoptimal result from [422].

Theorem 21.1. Let 1 < p < pS. Then the equation

ut −∆u = up, x ∈ Rn, t ∈ R (21.1)

has no positive, radial, bounded classical solution.

Theorem 21.1 is optimal in view of the existence of bounded positive radialstationary solutions for n ≥ 3 and p ≥ pS (see Section 9). It is very likely thatTheorem 21.1 should hold without the radial symmetry assumption, but this hasnot been proved so far. However, under the stronger restriction p < pB, where

pB :=

⎧⎨⎩∞ if n = 1,

n(n + 2)(n− 1)2

if n > 1,

we have the following Liouville-type theorem in the general (nonradial) case. It isa consequence of [79, Theorem 0.1].

21. Parabolic Liouville-type results 151

Theorem 21.2. Let 1 < p < pB. Then equation (21.1) has no positive classicalsolution.

Remark 21.3. Theorem 21.1 remains true for nontrivial nonnegative radial clas-sical solutions, bounded or not, whereas Theorem 21.2 remains true for nontrivialnonnegative classical solutions (see Remark 26.10(i) and cf. [425]).

The proofs of Theorems 21.1 and 21.2 are completely different, based on inter-section-comparison and integral estimates, respectively.

For the proof of Theorem 21.1, we need some simple preliminary observationsconcerning radial steady states. Let ψ1 be the solution of the equation

ψ′′ +n− 1

rψ′ + |ψ|p−1ψ = 0, r > 0, (21.2)

satisfying ψ(0) = 1, ψ′(0) = 0. Obviously ψ′′1 (0) < 0. It is known that the solution

is defined on some interval and it changes sign due to p < pS (this follows forinstance from Theorem 8.1). We denote by r1 > 0 its first zero. By uniqueness forthe initial-value problem, ψ′

1(r1) < 0. We thus have

ψ1(r) > 0 in [0, r1) and ψ1(r1) = 0 > ψ′1(r1).

Clearly, ψα(r) := αψ1(αp−12 r) is the solution of (21.2) with ψ(0) = α, ψ′(0) = 0,

and with the first positive zero rα = α− p−12 r1. As an elementary consequence of

the properties of ψ1 we obtain the following

Lemma 21.4. Given any m > 0, we have

limα→∞ (supψ′

α(r) : r ∈ [0, rα] is such that ψα(r) ≤ m) = −∞.

Proof of Theorem 21.1. The proof is by contradiction. Assume that u is apositive, bounded classical solution of (21.1), u(x, t) = U(r, t), where r = |x|.By the boundedness assumption and parabolic estimates, U and Ur are boundedon [0,∞) × R. It follows from Lemma 21.4 that if α is sufficiently large, thenU(·, t)− ψα has exactly one zero in [0, rα] for any t and the zero is simple.

We next claim that

z[0,rα](U(·, t)− ψα) ≥ 1 t ≤ 0, α > 0, (21.3)

where z[0,rα](w) denotes the zero number of the function w in the interval [0, rα](see Appendix F). Indeed, if not, then U(·, t0) > ψα in [0, rα] for some t0. ByTheorem 17.8 we know that each solution of the Dirichlet problem

ut −∆u = up, |x| < rα, t > 0,

u = 0, |x| = rα, t > 0,

u(x, t0) = U0(|x|), |x| < rα

⎫⎪⎬⎪⎭

blows up in finite time provided U0 > ψα in [0, rα). Choosing the initial functionU0 between ψα and U(·, t0) we conclude, by comparison, that u and u both blowup in finite time, in contradiction to the global existence assumption on u. Thisproves the claim.

Set

α0 := infβ > 0 : z[0,rα](U(·, t)− ψα) = 1 for all t ≤ 0 and α ≥ β.

In view of the above remark on large α, we have α0 < ∞. Also α0 > 0. Indeed, forsmall α > 0 we have ψα(0) < U(0, t) for t = 0 and for t > 0 small. By the propertiesof the zero number (see Theorem 52.28), we can choose t < 0 small such thatψα(0)−U(·, t) has only simple zeros and then, by (21.3), z[0,rα](U(·, t)−ψα) ≥ 2.

By definition of α0 (and (21.3)), there are sequences αk α0 and tk ≤ 0 suchthat

z[0,rαk](U(·, tk)− ψαk

) ≥ 2, k = 1, 2, . . . .

Using Theorem 52.28 again, we get

z[0,rαk](U(·, tk + t)− ψαk

) ≥ 2, t ≤ 0, k = 1, 2, . . . . (21.4)

This in particular allows us to assume, choosing different tk if necessary, thattk → −∞. By the boundedness assumption and parabolic estimates, passing to asubsequence, we may further assume that

u(x, tk + t)→ v(x, t), x ∈ Rn, t ∈ R,

with convergence in C2,1(Rn × R). Clearly then, there is δ > 0 such that for eachfixed t,

U(·, tk + t)− ψαk→ V (·, t)− ψα0

in C1[0, rα0 + δ], where v(x, t) = V (|x|, t). This and (21.4) guarantee that foreach t ≤ 0, V (·, t) − ψα0 has at least two zeros or a multiple zero in [0, rα0). Bythe properties of the zero number (see Theorem 52.28), we may choose t < 0 sothat V (·, t) − ψα0 has only simple zeros (and, hence at least two of them). SinceU(·, tk+t)−ψα0 is close to V (·, t)−ψα0 in C1[0, rα0 ], if k is large, it has at least twosimple zeros in [0, rα0) as well. But then, for α > α0, α close to α0, the functionu(·, tk + t)−ψα has at least two zeros in [0, rα), contradicting the definition of α0.

We have thus shown that the assumption u ≡ 0 leads to a contradiction, whichproves the theorem.

We now turn to the proof of Theorem 21.2. It will be a direct consequence ofthe following space-time integral estimates [79] for (local) solutions of (21.1).

Proposition 21.5. Let 1 (n+2)(p− 1)/2 such that if 0 < u ∈ C2,1(B1× (−1, 1)) is a solutionof

ut −∆u = up, |x| < 1, −1 < t < 1,

then ∫ 1/2

−1/2

∫|x|<1/2

ur dx dt ≤ C(n, p).

Let us first prove Theorem 21.2 assuming Proposition 21.5. It suffices to applya simple homogeneity argument.

Proof of Theorem 21.2. Let R > 0. Let u be a solution of (21.1). Then, foreach R > 0, v(x, t) := R2/(p−1)u(Rx, R2t) solves (21.1) in B1 × (−1, 1). It followsfrom Proposition 21.5 that∫ R2/2

−R2/2

∫|y|<R/2

ur(y, s) dy ds = Rn+2

∫ 1/2

−1/2

∫|x|<1/2

ur(Rx, R2t) dx dt

= Rn+2−2r/(p−1)

∫ 1/2

−1/2

∫|x|<1/2

vr(x, t) dx dt ≤ C(n, p)Rn+2−2r/(p−1).

Since r > (n+2)(p−1)/2, by letting R →∞, we conclude that∫∞−∞

∫Rn ur dy ds =

0, hence u ≡ 0.

The proof of Proposition 21.5 uses the following key gradient estimate, which isthe analogue of the one used in Section 8 to prove the Liouville-type theorem andthe local estimates for the elliptic equation −∆u = up. In the rest of this section,we use the notation

∫ ∫=∫ T

−T

∫Ω

for simplicity.

Lemma 21.6. (i) Let Ω be an arbitrary domain in Rn, T > 0, and 0 ≤ ϕ ∈

D(Ω × (−T, T )). Let 0 < u ∈ C2,1(Ω × (−T, T )), be a solution of (21.1) in Ω ×(−T, T ). Fix k ∈ R with k = −1 and denote

I =∫ ∫

ϕu−2|∇u|4, L =∫ ∫

ϕu2p,

where, here and below, integrals are over Ω× (−T, T ). Then there holds

αI + δL ≤ C(n, p, k)∫ ∫

ϕ[(ut)2 + |ut|u−1|∇u|2

]+ |∇u|2|∆ϕ|

+ C(n, p, k)∫ ∫

(up + |ut|+ u−1|∇u|2)|∇u · ∇ϕ|+ up+1|ϕt|,(21.5)

where

α = −((n− 1)k + n)k

n, δ = −n− 1 + (n + 2)k/p

n. (21.6)

Assume 1 0. (21.7)

The main ingredient in the proof of Lemma 21.6 is Lemma 8.9, proved in Sec-tion 8, which provides a family of integral estimates relating any C2-function withits gradient and its Laplacian.

Proof. (i) We apply Lemma 8.9 with q = 0. Denoting

J =∫ ∫

ϕu−1|∇u|2∆u, K =∫ ∫

ϕ (∆u)2,

this gives us

−(n− 1

nk + 1

)kI +

n + 2n

kJ − n− 1n

K

≤ 12

∫ ∫|∇u|2∆ϕ +

∫ ∫ [∆u− ku−1|∇u|2

]∇u · ∇ϕ.

(21.8)

Now, since ∆u = ut − up, integrating by parts in t and/or in x, we obtain

K =∫ ∫

ϕ (ut)2 +∫ ∫

ϕu2p − 2∫ ∫

ϕuput

=∫ ∫

ϕ (ut)2 + L +2

p + 1

∫ ∫up+1ϕt

and

pJ = −∫ ∫

ϕ∇u · ∇(up) + p

∫ ∫ϕut u−1|∇u|2

=∫ ∫

ϕ (∆u)up +∫ ∫

(∇ϕ · ∇u)up + p

∫ ∫ϕut u−1|∇u|2

= −L− 1p + 1

∫ ∫up+1ϕt +

∫ ∫(∇ϕ · ∇u)up + p

∫ ∫ϕut u−1|∇u|2.

Substituting in (21.8), we obtain (21.5).(ii) For k < 0, the condition α, δ > 0 is equivalent to

(n− 1)p/(n + 2) < −k < n/(n− 1).

Such choice of k < 0 is clearly possible if p < pB.

Proof of Proposition 21.5. Taking k as in Lemma 21.6(ii), we shall estimatethe terms on the RHS of (21.5). Let us first prepare a suitable test-function. We


take ξ ∈ D(B1 × (−1, 1)), such that ξ = 1 in B1/2 × (−1/2, 1/2) and 0 ≤ ξ ≤ 1.By taking ϕ = ξ4p/(p−1), we have

|∇ϕ| ≤ Cϕ(3p+1)/4p, |∆ϕ| ≤ Cϕ(p+1)/2p, |ϕt| ≤ Cϕ(3p+1)/4p ≤ Cϕ(p+1)/2p.(21.9)

We first observe that∫ ∫|∇u|2

(|∆ϕ|+ ϕ−1|∇ϕ|2 + |ϕt|

)≤ η(I + L) + C(η), η > 0. (21.10)

Indeed, this follows from Young’s inequality and (21.9), by writing

|∇u|2(|∆ϕ|+ ϕ−1|∇ϕ|2 + |ϕt|

)≤ ηϕu−2|∇u|4 + C(η)ϕ−1u2(|∆ϕ|+ ϕ−1|∇ϕ|2 + |ϕt|

)2≤ ηϕu−2|∇u|4 + C(η)ϕ1/pu2

≤ ηϕu−2|∇u|4 + ηϕu2p + C(η).

Now fix ε > 0. Using Young’s inequality, (21.9) and (21.10), we estimate the RHSof (21.5) as follows:∫ ∫

ϕ[(ut)2 + |ut|u−1|∇u|2

]+ |∇u|2|∆ϕ|

+∫ ∫

(up + |ut|+ u−1|∇u|2)|∇u · ∇ϕ|+ up+1|ϕt|

≤ ε

∫ ∫ϕ[u2p + u−2|∇u|4

]+ C(ε)

∫ ∫ [ϕ(ut)2 + |∇u|2(ϕ−1|∇ϕ|2 + |∆ϕ|) + (ϕ−(p+1)|ϕt|2p)1/(p−1)

]≤ 2ε(I + L) + C(ε)

(1 +

∫ ∫ϕ(ut)2

).

(21.11)Let us handle the last term in the above inequality. Multiplying equation (21.1)by ϕut, integrating by parts in x and t, and using Young’s inequality and (21.9),we get, for each η > 0,∫ ∫

ϕ (ut)2 =∫ ∫

ϕ∂t

( up+1

p + 1− |∇u|2

2

)− (∇ϕ · ∇u)ut

=∫ ∫ ( |∇u|2

2− up+1

p + 1

)ϕt − (∇ϕ · ∇u)ut

≤ 12

∫ ∫|∇u|2

(|ϕt|+ |∇ϕ|2ϕ−1

)+

12

∫ ∫ϕ (ut)2 +

1p + 1

∫ ∫up+1|ϕt|.

By (21.10) and (21.9), for all η > 0, it follows that∫ ∫ϕ (ut)2 ≤

∫ ∫|∇u|2

(|ϕt|+ |∇ϕ|2ϕ−1

)+

2p + 1

∫ ∫up+1|ϕt|

≤ η(I + L) + C(η) + η

∫ ∫ϕu2p + C(η)

∫ ∫ϕ−(p+1)/(p−1)|ϕt|2p/(p−1)

≤ 2η(I + L) + C(η).(21.12)

Combining (21.12), applied with η = ε(2C(ε))−1, (21.11) and (21.5), we obtain

αI + δL ≤ C(n, p)ε(I + L) + C(ε).

Since α, δ > 0, by choosing ε = ε(n, p) sufficiently small, we conclude that I, L ≤C.

Remark 21.7. In the above proof, it is a priori possible to use Lemma 8.9 withvalues other than q = 0 (at the expense of additional complications in the estimateof the terms on the RHS of (21.5)). However, this does not seem to enable one togo beyond the condition p < pB.

We now consider the case of a half-space. The following result was proved in[425].

Theorem 21.8. Let p > 1. Assume n ≤ 2, or p < (n − 1)(n + 1)/(n − 2)2 andn ≥ 3. Then the problem

ut −∆u = up, x ∈ Rn+, t ∈ R,

u = 0, x ∈ ∂Rn+, t ∈ R

(21.13)

has no positive bounded classical solution.

Remarks 21.9. (a) We note that (n− 1)(n + 1)/(n− 2)2 is the exponent pB ofTheorem 21.2 in dimension n− 1, and that this number is greater than pS .

(b) Any nontrivial bounded classical solution u ≥ 0 of (21.13) is positive (thisfollows from the argument after (26.44) in the proof of Theorem 26.8). On the otherhand, it can be shown [425] that if p < pB = n(n + 2)/(n − 1)2, then problem(21.13) has no nontrivial nonnegative classical solution, bounded or not.

Theorem 21.8 is a consequence of Theorem 21.2 and the following monotonicityresult [425] concerning the more general problem

ut −∆u = f(u), x ∈ Rn+, t ∈ R,

u = 0, x ∈ ∂Rn+, t ∈ R,

(21.14)

where f is a C1-function.

Theorem 21.10. Assume f : [0,∞) → R is a C1-function satisfying f(0) = 0and f ′(0) ≤ 0. Then the following statements hold true.(i) Each positive bounded solution u of (21.14) is increasing in x1:

∂x1u(x, t) > 0, x ∈ Rn+, t ∈ R.

(ii) If there is a positive bounded solution of (21.14), then there exists a positivebounded solution of

ut −∆u = f(u), x ∈ Rn−1, t ∈ R. (21.15)

For n = 1, equation (21.15) should be understood as the ordinary differentialequation ut = f(u).

The proofs of both statements (i) and (ii) use extensions of moving plane argu-ments of [150] to parabolic equations. A straightforward modification of the proofbelow shows that (i), (ii) hold for positive bounded solutions defined on (−∞, T )for some T > 0.

Proof. First we prove (i). We use the following notation. For λ > 0 let

Tλ = x ∈ Rn : 0 < x1 < λ.

For a function z defined on Rn+ let zλ and Vλz be functions on Tλ defined by

zλ(x) = z(2λ− x1, x′),

Vλz(x) = zλ(x) − z(x),

(21.16)

where x′ = (x2, x3, . . . , xn).Let u be a positive bounded solution of (21.14). Observe that for each λ > 0,

v = Vλu satisfies

vt −∆v = cλ(x, t)v, x ∈ Tλ, t ∈ R,

v = 0, x1 = λ, x′ ∈ Rn−1, t ∈ R,

v > 0, x1 = 0, x′ ∈ Rn−1, t ∈ R,

⎫⎪⎬⎪⎭ (21.17)

where

cλ(x, t) =∫ 1

0

f ′(u(x, t) + s(uλ(x, t)− u(x, t))) ds. (21.18)

Our goal is to prove that the statement

Vλu(x, t) ≥ 0 (x ∈ Tλ, t ∈ R) (S)λ

holds for each λ > 0. Once this is done, the maximum principle applied to theabove linear problem guarantees that we have in fact the strict inequality in (S)λ

and the Hopf boundary principle then gives

2∂x1u(x, t)

x1=λ= −∂x1Vλu(x, t)

x1=λ

> 0

for each λ > 0, proving (i).We shall use the following lemma [150].

Lemma 21.11. Given any positive constants q, λ satisfying λ−2π2 > q, thereexists a smooth function h on Tλ such that

∆h + qh = 0, x ∈ Tλ,

h(x) > 0, x ∈ Tλ,

h(x) →∞, |x| → ∞, x ∈ Tλ.

⎫⎪⎬⎪⎭ (21.19)

Moreover, h satisfies h ≥ η for some constant η > 0.

Proof. A straightforward computation shows that

h = h(x1, x2, . . . , xn) = cos[π(2x1 − λ)

2(λ + ε)

] n∏i=2

cosh(εxi)

satisfies the required properties for ε > 0 small.

We first prove that (S)λ holds for λ small. Fix a positive constant γ and set

q := supt∈R, x∈Rn

+

f ′(u(x, t)) + γ. (21.20)

If λ > 0 is sufficiently small, so that λ−2π2 > q, we can apply Lemma 21.11. Withthe resulting function h, we consider the problem satisfied by w := eγtv/h, wherev = Vλu. A simple computation using (21.17), (21.19) shows that

wt −∆w − 2∇h

h· ∇w − (γ + cλ(x, t)− q)w = 0, x ∈ Tλ, t ∈ R,

w ≥ 0, x ∈ ∂Tλ, t ∈ R,

w(x, t) → 0, |x| → ∞, x ∈ Tλ, t ∈ R.

⎫⎪⎪⎬⎪⎪⎭ (21.21)

The choice of q implies γ + cλ− q ≤ 0 in Tλ×R. Applying the maximum principleon Tλ × (t0, t), for each t0 < t, we obtain

supx∈Tλ

w−(x, t) ≤ supx∈Tλ

w−(x, t0). (21.22)

For v the above inequality means

supx∈Tλ

v−(x, t)h(x)

≤ e−γ(t−t0) supx∈Tλ

v−(x, t0)h(x)

. (21.23)

In view of boundedness of v = Vλu, letting t0 → −∞ we obtain that v ≥ 0everywhere. Using the maximum principle again we conclude that v is positive inTλ × R, hence (S)λ holds.

In the next step we denote

λ0 = supµ > 0 : (S)λ holds for all λ ∈ (0, µ). (21.24)

As proved above, λ0 > 0. We now show by contradiction that λ0 = ∞. Assumeλ0 < ∞. Then there is a sequence λk ≥ λ0 such that λk → λ0 and the set

Zk := (x, t) ∈ Tλk× R : Vλk

u(x, t) < 0

is nonempty. Set

mk := supu(y1, x′, t) : y1 ∈ (0, λk), x′ ∈ R

n−1, t ∈ R, and

there exists x1 ∈ (0, λk) such that (x1, x′, t) ∈ Zk.

We consider the following two possibilities.(a) mk → 0,(b) passing to a subsequence we have mk ≥ ε0 for some ε0 > 0.

First assume that (b) holds. Then there are sequences xk1 , yk

1 ∈ (0, λk), zk ∈R

n−1, tk ∈ R such that Vλku(xk

1 , zk, tk) < 0 and u(yk1 , zk, tk) ≥ ε0. We may

assume that xk1 → a and yk

1 → b for some a, b ∈ [0, λ0] . Consider the functions

uk(x, t) := u(x1, x′ + zk, t + tk), x = (x1, x

′) ∈ Rn, t ∈ R.

Each of them is a positive solution of (21.14) satisfying Vλkuk(xk

1 , 0, 0) < 0,uk(yk

1 , 0, 0) ≥ ε0 and Vλ0uk ≥ 0 in Tλ0 × R (the last inequality follows from thedefinition of λ0 and continuity). Moreover, the sequence uk is uniformly bounded.Using standard parabolic estimates, one shows that if uk is replaced by a subse-quence, then it converges in C2,1(Rn+1) to a nonnegative solution u of (21.14).The above properties of uk imply Vλ0 u(a, 0, 0) ≤ 0, u(b, 0, 0) ≥ ε0, and Vλ0 u ≥ 0in Tλ0 × R. Since u is nontrivial and f(0) = 0 the maximum principle guaranteesthat u is positive everywhere. Consequently, v := Vλ0 u solves the correspondingproblem (21.17) with λ = λ0 and therefore v > 0 in Tλ0×R. It follows in particularthat necessarily a = λ0. By the Hopf principle,

2ux1(λ0, 0, 0) = −∂x1Vλ0 u(x1, 0, 0)

x1=λ0> 0.

Consequently, ux1(x1, 0, 0) is bounded below by a positive constant on an intervalaround λ0 and this remains valid if u is replaced by uk for k large. That is, thereis δ > 0 such that

∂x1u(x1, zk, tk) = ∂x1uk(x1, 0, 0) > 0, x1 ∈ [λ0 − δ, λ0 + δ], (21.25)

for all sufficiently large k. However, since 2λk−xk1 > xk

1 both belong to [λ0−δ, λ0+δ]for large k, (21.25) contradicts the assumption that Vλk

u(xk1 , zk, tk) < 0.

We have shown that (b) leads to a contradiction. Assume now that (a) holds.Consider problem (21.17) with λ = λk and k sufficiently large. We are goingto apply the maximum principle on the set Zk (assumed to be nonempty). Theboundary conditions in (21.17) imply v = Vλk

u = 0 on ∂Zk. Next observe thatproperty (a), in conjunction with (21.18) and the definition of mk, guarantees thatfor

qk := sup(x,t)∈Zk

cλk(x, t)

we havelim sup

k→∞qk ≤ 0.

Fix k so large that q := qk + γ < λ−2k π2, where γ is some positive constant, and

set λ = λk. Apply Lemma 21.11 and let h be the resulting function. As in ourarguments above, w := eγtv/h satisfies problem (21.21). This time we know thatγ + cλ − q ≤ 0 on Zk only. However, since v vanishes on ∂Zk, we can still applythe maximum principle on Zk to conclude that (21.22) holds and, consequently,that v ≥ 0 in Zk. This of course contradicts the definition of Zk. Thus possibility(a) leads to a contradiction, too, which proves that λ0 = ∞.

We have completed the proof of assertion (i).To prove assertion (ii), let u be a positive bounded solution of (21.14). For

k = 1, 2, . . . consider the functions

uk(x1, x′, t) := u(x1 + k, x′, t), (x1, x

′, t) ∈ (−k,∞)× Rn−1 × R.

Each of them solves the equation ut−∆u = f(u) on its domain. Since the sequenceis uniformly bounded, using parabolic estimates one shows that a subsequence ofuk converges uniformly on each compact to a bounded nonnegative solution u ofut−∆u = f(u) on R

n×R. From the monotonicity of u proved in (c1), we furtherconclude that u is positive and independent of x1. This proves assertion (ii).

Remark 21.12. Liouville-type result under a decay assumption at −∞.A different parabolic Liouville-type theorem was proved in [367] for 1 < p < pS .Namely, if u is a classical solution of

ut −∆u = |u|p−1u (21.26)

on Rn × (−∞, 0) and is such that

supt<0

|t|1/(p−1)‖u(t)‖∞ <∞, (21.27)

then u depends only on t. This result was used in [367] to obtain refined blow-upestimates for problem (18.1) near the blow-up time. It implies in particular that(21.1) has no positive bounded classical solution satisfying (21.27) for 1 < p < pS .However, it does not seem possible to use this form of Liouville-type theorem toestablish universal blow-up estimates on the whole existence interval (0, T ), likethose which will be derived from Theorems 21.1 and 21.2 in Section 21.

22. A priori bounds 161

22. A priori bounds

Consider the model problem

ut −∆u = |u|p−1u, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (22.1)

where Ω is bounded and p > 1. We have seen that (22.1) admits both:• finite-time blow-up solutions — cf. Section 17; and• global bounded solutions (in particular small data solutions decaying to 0 as

t→∞, and stationary solutions if p < pS) — cf. Sections 19 and 6.In order to understand the structure of solutions of problem (22.1), it is naturalto investigate whether or not it admits other kinds of solutions (namely globalunbounded classical solutions). In the case when all global solutions are bounded,one can further look for an a priori estimate of global solutions, that is, anestimate of the form

supt≥0

‖u(t)‖∞ ≤ C(‖u0‖∞), with C bounded on bounded sets. (22.2)

This estimate means that, given K > 0, there exists C = C(K) > 0 such thatall global solutions with ‖u0‖∞ ≤ K satisfy ‖u(t)‖∞ ≤ C for all t ≥ 0. Theexistence of stronger universal bounds (independent of initial data) will be studiedin Section 26.

We shall see that the answers to these questions (boundedness of global solutionsvs. existence of unbounded global solutions, existence vs. nonexistence of a prioriestimates) strongly depend on the value of p. Besides the intrinsic interest of suchquestions, let us emphasize that the results and techniques of proofs have manyapplications (see Section 28 and cf. also, for instance, Theorem 22.13, the proof ofTheorem 23.7, Remark 23.14, and the proof of Theorem 27.2).

22.1. A priori bounds in the subcritical case

In this subsection we establish a priori estimates of global solutions in the subcrit-ical case p < pS . As we shall see below, the assumption p < pS is necessary for thebound (22.2) (at least if Ω is starshaped).

Theorem 22.1. Assume Ω bounded and 1 < p < pS. Then the bound (22.2) istrue for all global solutions of (22.1).

This result was proved in [243] for u0 ≥ 0 and in [437] in the general case.Earlier partial results in that direction can be found in [403], [396], [114], [186].

We shall first prove the above theorem under the additional assumption u0 ≥ 0.This proof is due to [243] and it is based on rescaling arguments (similar to thoseused in the proof of Theorem 12.1) and on the energy functional E.

Proof of Theorem 22.1 for nonnegative solutions. Assume that the bound(22.2) does not hold for global nonnegative solutions. Then there exist tk > 0 andu0,k ≥ 0 such that ‖u0,k‖∞ ≤ C0 and the solutions uk := u(·; u0,k) satisfy

Mk := uk(xk, tk) = supuk(x, t) : x ∈ Ω, t ∈ [0, tk] → ∞ as k →∞. (22.3)

Let ψ be the solution of ψ(0) = C0, ψ′(t) = ψp(t) for t > 0, and let δ = δ(C0, p) > 0be such that ψ(δ) = 2C0. Then the comparison principle shows uk(x, t) ≤ ψ(t) ≤2C0 for all x ∈ Ω and t ∈ [0, δ], hence tk ≥ δ for k large enough. Now the variation-of-constants formula (15.5) and the estimate

‖e−tAw‖1,2 ≤ C1t−1/2‖w‖2 ≤ C2t

−1/2‖w‖∞

easily imply ‖uk(δ/2)‖1,2 ≤ C, where by C we denote a positive constant whichdoes not depend on k. This estimate and Theorem 17.6 guarantee

0 ≤ E(uk(δ/2)

)< C. (22.4)

Denote νk := M−(p−1)/2k and set

vk(y, s) :=1

Mkuk(xk + νky, tk + ν2

ks), (y, s) ∈ Qk,

where Qk := (y, s) : (xk + νky, tk + ν2ks) ∈ Ω× (0, tk). Then 0 ≤ vk(y, s) ≤ 1 =

vk(0, 0) and vk solves the problem

∂svk −∆yvk = vpk in Qk,

vk = 0 for (y, s) ∈ ∂Qk, − tkν2

k

< s < 0.

Denote dk := dist (xk, ∂Ω). Passing to a subsequence we may assume that one ofthe following cases occurs: (i) dk/νk →∞, (ii) dk/νk → c ≥ 0.

Case (i). Set

Qk := (y, s) : |y| < dk

νk, − tk

2ν2k

< s < 0.

Then Qk ⊂ Qk and the parabolic Lp-estimates (see Appendix B) together withstandard embedding theorems guarantee the boundedness of vk in the spaceCα,α/2(Rn × (−∞, 0)) for some α > 0. Consequently, given β ∈ (0, α), we mayassume vk → v in Cβ,β/2(Rn × (−∞, 0)), where v is a classical solution of

vs −∆v = vp in Rn × (−∞, 0) (22.5)

satisfying 0 ≤ v ≤ v(0, 0) = 1. Now setting σ := 4/(p− 1)− (n− 2) > 0 and using(22.4) we obtain∫∫

Qk

|∂svk|2 dy ds = νσk

∫ tk

tk/2

∫|x−xk|<dk

|∂tuk|2 dx dt ≤ νσk

∫ ∞

δ/2

∫Ω

|∂tuk|2 dx dt

≤ νσk

[E(uk(δ/2)

)− lim

t→∞E(uk(t)

)]→ 0.

Since ∂svk → vs in D′(Rn×(−∞, 0)), it follows that vs ≡ 0. Now (22.5) contradictsTheorem 8.1.

Case (ii). In this case we obtain, similarly as in Case (i), a function v solvingthe problem

vs −∆v = vp in Hnc × (−∞, 0),

v = 0 on ∂Hnc × (−∞, 0),

(22.6)

and satisfying 0 ≤ v ≤ v(0, 0) = 1, where Hnc := y ∈ R

n : y1 > −c (see [243] fordetails and cf. also the proof of Theorem 12.1). As in Case (i) we obtain vs ≡ 0,hence (22.6) contradicts Theorem 8.2.

Now we are going to prove Theorem 22.1 in the general case. The proof is basedon energy estimates, interpolation, maximal regularity, and a bootstrap argument.The first two ingredients were first used in [114], where the authors had to assumep(3n − 4) < (3n + 8). The bootstrap argument (which enables one to get rid ofthis additional assumption on p) appeared for the first time in [437].

Proof of Theorem 22.1. Let M > 0 and let u be a global solution of (22.1)with ‖u0‖∞ ≤ M . We shall denote by C, C1, C2 various positive constants whichdepend on u0 through M only and which may vary from step to step. Also, by theword “bounded”, we mean that the bound depends on u0 through M only.

As in the proof of Theorem 22.1 for nonnegative solutions, there exists δ =δ(M) > 0 such that ‖u(t)‖∞ ≤ C for t ∈ [0, δ] and ‖u(δ)‖1,2 ≤ C. Hence we mayassume ‖u0‖1,2 ≤ C. Since u is global, Theorem 17.6 and Remark 17.7 guarantee

0 ≤ E(u(t)

)≤ C, t ≥ 0, (22.7)

and‖u(t)‖2 ≤ C, t ≥ 0. (22.8)

Consequently, ∫ ∞

0

∫Ω

u2t dx dt = E(u0)− lim

t→∞ E(u(t)

)≤ C. (22.9)

This estimate and (22.8) guarantee that

u is bounded in W 1,2([t, t + 1], L2(Ω)

)uniformly for t ≥ 0. (22.10)

Multiplying the equation in (22.1) by u we get∫Ω

uut dx = −∫

Ω

|∇u(t)|2 dx +∫

Ω

|u(t)|p+1 dx = −2E(u(t)

)+C

∫Ω

|u(t)|p+1 dx,

so that, for each r ≥ 1, (22.7) implies∫ t+1

t

(∫Ω

|u|p+1 dx)r

ds ≤ C[1 +

∫ t+1

t

(∫Ω

|uut| dx)r

ds], t ≥ 0. (22.11)

Notice that Cauchy’s inequality, (22.8) and (22.9) imply∫ t+1

t

(∫Ω

|uut| dx)2

ds ≤∫ t+1

t

(∫Ω

u2 dx)(∫

Ω

u2t dx

)ds ≤ C,

hence we infer from (22.11) that

u is bounded in L(p+1)r([t, t + 1], Lp+1(Ω)

)uniformly for t ≥ 0, (22.12)

if r = 2. Now (22.10), (22.12) and (51.6) guarantee

‖u(t)‖q ≤ Cq for all t ≥ 0 and q < qr := p + 1− p− 1r + 1

, (22.13)

where r = 2. Theorem 15.2 or Remark 51.37(iii) (see also Theorem 16.4) implyour assertion provided supt≥0 ‖u(t)‖q ≤ C for some q > n(p− 1)/2. This estimatefollows from (22.13) if

n

2(p− 1) < p + 1− p− 1

r + 1. (22.14)

If r = 2, then (22.14) is equivalent to p(3n− 4) < 3n + 8 (which is the conditionof [114]). In what follows we shall use a bootstrap argument to show that (22.13)is true for any r ≥ 2. Since (22.14) reduces to p < pS if r →∞ we shall be done.

We already know (see the beginning of the proof) that there exists δ = δ(M) > 0such that ‖u(t)‖∞ ≤ C for t ∈ [0, δ]. We claim that for any interval I ⊂ [0,∞) oflength δ there exists τ ∈ I such that ‖u(τ)‖BC2 ≤ C. In fact, let I = (t, t + δ) andset J := (t, t + δ/2). Then (22.7) and (22.12) with r = 2 imply∫

J

(∫Ω

|∇u|2 dx)2

ds ≤ C[1 +

∫J

(∫Ω

|u|p+1 dx)2

ds]≤ C,

hence there exist C1 > 0 and τJ ∈ J such that ‖u(τJ)‖1,2 ≤ C1. The well-posednessof (22.1) in W 1,2

0 (Ω) (see Example 51.10 and Theorem 51.7) guarantees the exis-tence of η = η(C1) > 0 and C2 = C2(C1) > 0 such that η < δ/2 and ‖u(s)‖1,2 ≤ C2

for all s ∈ [τJ , τJ + η]. Now standard regularity results (see Example 51.27 and

Appendix B) guarantee ‖u(τJ + η)‖BC2 ≤ C, where C = C(η, C2). Hence it issufficient to put τ := τJ + η.

Next assume that r ≥ 2 and∫ t+1

t

(∫Ω

|u|p+1 dx)r

ds ≤ C for all t ≥ δ. (22.15)

We shall show that the same estimate is true with r replaced by r for any r ∈(r, r + 2). Since (22.15) is true for r = 2, an obvious bootstrap argument willguarantee (22.15) for any r ≥ 2. Since (22.15) implies (22.13), the conclusion willfollow.

Hence let r ∈ (r, r + 2), and consider q < qr (q close to qr). Set

p := (p + 1)/p, θ :=p + 1p− 1

q − 2q

∈ (0, 1), β := 2/(r(1− θ)) > 1.

Choose t ≥ δ and τ ∈ (t − δ, t) such that ‖u(τ)‖BC2 ≤ C. Using successively(22.11), Holder’s inequality and (22.13), interpolation, Holder’s inequality, (22.9),the maximal regularity property (51.8), and ‖u(τ)‖BC2 ≤ C, we obtain

∫ t+1

τ

(∫Ω

|u|p+1 dx)r

ds ≤ C[1 +

∫ t+1

τ

‖uut‖r1 ds]

≤ C[1 +

∫ t+1

τ

‖ut‖rq′ ds

]≤ C

[1 +

∫ t+1

τ

‖ut‖rθp ‖ut‖r(1−θ)

2 ds]

≤ C[1 +

(∫ t+1

τ

‖ut‖rθβ′p ds

)1/β′(∫ t+1

τ

‖ut‖22 ds)1/β]

≤ C[1 + ‖u(τ)‖BC2 +

(∫ t+1

τ

‖|u|p−1u‖rθβ′p ds

)1/β′]≤ C

[1 +

(∫ t+1

τ

(∫Ω

|u|p+1 dx)rθβ′p/(p+1)

ds)1/β′]

.

Since r < r+2 we can choose q close to qr so that rθβ′p/(p+1) < r. Consequently,(22.15) and the last estimate guarantee (22.15) with r replaced by r.

Remarks 22.2. Uniform bound in terms of the energy. Let Ω be bounded,p < pS , u be a solution of (22.1) on the time interval [0, T ), T < ∞ and M > 0.

(i) If

‖u0‖∞ ≤M, E(u(t)) ≥ −M and ‖u(t)‖2 ≤ M, t ∈ [0, T ),

then‖u(t)‖∞ ≤ C(M), t ∈ [0, T ).

This follows from the above proof of Theorem 22.1 by replacing the interval [0,∞)with [0, T ).

(ii) If‖u0‖∞ ≤ M and E(u(t)) ≥ −M, t ∈ [0, T ), (22.16)

then‖u(t)‖∞ ≤ C(M, T ), t ∈ [0, T ).

In fact, (22.16), (17.10) and Gronwall’s inequality guarantee

‖u(t)‖2 ≤ C(K, M, T ), t < T,

where K stands for a bound on ‖u0‖2. Therefore the assertion follows from (i).

Remark 22.3. Cauchy problem. Let Ω = Rn and 1 < p < pS . Then (22.2)

is still true for positive radial solutions (and for all positive solutions providedp < pB), see Theorem 26.9 below. Using the same approach as in the proof ofTheorem 22.1 for nonnegative solutions one can also show weaker estimate

‖u(t)‖∞ ≤ C(‖u0‖∞, E(u0)) for all t ≥ 0

for nonnegative initial data u0 ∈ H1(Rn).If we consider problem (17.1) with λ < 0, 1 < p < pS , Ω = R

n and initialdata in X := L∞ ∩ L(p+1)/p ∩H1(Rn), then any global (not necessarily positive)solution satisfies the estimate

‖u(t)‖X ≤ C(‖u0‖X) for all t ≥ 0,

see [440]. The same result remains true for X := H1(Rn) or X := L∞ ∩H1(Rn)due to a recent result in [537].

22.2. Boundedness of global solutions in the supercriticalcase

Consider problem (22.1), where Ω is a ball and u0 ∈ L∞(Ω) is a nonnegative radialfunction. If the solution u is global and p < pS , then Theorem 22.1 guarantees theboundedness of u, i.e.:

supt≥0

‖u(t)‖∞ < ∞. (22.17)

In this subsection we show that this property remains true for p > pS .Let us emphasize that the bound (22.17) does not imply the stronger a priori

estimate (22.2). In fact, we will see in Theorem 28.7(iv) that estimate (22.2) failswhenever p ≥ pS .

Theorem 22.4. Assume p > pS, Ω = BR, and let u0 ∈ L∞(Ω), u0 ≥ 0, be radial.If the solution u of (22.1) is global, then property (22.17) is true.

We will prove Theorem 22.4 only under the additional assumption p < pL,where

pL := ∞ if n ≤ 10,

1 + 6n−10 if n > 10.

(22.18)

Notice that pL > pJL if n > 10, where pJL is defined in (9.3). If n > 10 and p >pJL, then the statement of Theorem 22.4 follows from [378]. See also Remark 23.13for an alternative proof, due to [125], in the case p < pJL.

In the proof of the above theorem we will need the following result.

Proposition 22.5. Let pS 0,

ϕ′(0) = 0,

satisfying limy→∞ ϕ(y)y2/(p−1) = B ∈ (0, cp).Given T ∈ R, set

w(r, t) := (T − t)−1/(p−1)ϕ(r/√

T − t)

for r ≥ 0, t < T,

w(r, T ) := limt→T+

w(r, t) for r > 0.

Thenwt − wrr −

n− 1r

wr = wp, r > 0, t < T,

w(r, T ) = Br−2/(p−1), r > 0.

The function w in the preceding proposition is a backward self-similar solu-tion of problem (22.1). Proposition 22.5 follows from [105], [325] (if p < pJL) and[326] (if p ≥ pJL). Since the corresponding proofs are quite long, we will prove itjust in the case p = 2 when one can find an explicit formula for ϕ (due to [225]).Let us note that in the case p < pJL there exist infinitely many functions ϕ withthe required properties, and that both numerical and analytic results indicate thatsuch solutions do not exist if p > pL, see [418], [376].

Proof of Proposition 22.5 for p = 2. Let p = 2 and 6 < n < 16 (this corre-sponds to pS < 2 < pL). Set

ϕ(y) :=A

(a + y2)2+

B

a + y2,

where

A := 48(10D− (n + 14)), B := 24(D − 2), D :=√

1 + n/2.

It is easy to see that ϕ possesses the required properties. In particular, B < c2 =2(n− 4).

The following proof is due to [232].

Proof of Theorem 22.4 for p < pL. Let U∗(r) = cpr−2/(p−1) be the singular

solution defined in (3.9). Assume on the contrary that u is a global unboundedclassical solution. Since u is radial (see Remark 16.2(i)), we have u(x, t) = U(|x|, t)for some U : [0, R]× (0,∞)→ R.

0 R

U∗(δ)

U∗

Uδ

U(·, t1)

Figure 9: Graphs of U∗, Uδ, U(·, t1) if z(U(·, t0)− U∗) = 0.

Assume z(U(·, t0) − U∗) ≤ 1 for some t0 > 0, where z(ψ) denotes the zeronumber of the function ψ in the interval (0, R) (see Appendix F). Since U(0, t0) <U∗(0) = ∞ and 0 = U(R, t0) < U∗(R) we have z(U(·, t0)− U∗) = 0. ConsequentlyU(·, t0) ≤ U∗. Fix t1 > t0. Then by the maximum principle there exists ε > 0 suchthat U(·, t1) ≤ U∗ − ε and we may find δ > 0 such that the function Uδ(r) :=U∗(r + δ) lies above U(·, t1). Since

−U ′′δ −

n− 1r

U ′δ ≥ Up

δ , 0 < r < R,

with −U ′δ(0) > 0 and Uδ(R) > 0, it follows from the maximum principle that

U(r, t) ≤ Uδ(r) ≤ U∗(δ) for all r ∈ [0, R] and t ≥ t1, see Figure 9. However, thiscontradicts our assumptions. Consequently,

z(U(·, t)− U∗) ≥ 2 for all t > 0. (22.19)

Fix τ > 0 small. Since U(r, τ) > 0 for r ∈ [0, R) and Ur(R, τ) < 0 by themaximum principle, we can find T large enough such that the backward self-similarsolution w from Proposition 22.5 satisfies z(U(·, τ)− w(·, τ)) = 1, see Figure 10.

0 R

w(·, τ)

U(·, τ)

Figure 10: Graphs of U(·, τ), w(·, τ).

Consequently, Theorem 52.28 implies

z(U(·, t)− w(·, t)) ≤ 1 for all t ∈ [τ, T ). (22.20)

However, w(·, T ) < U∗ so that (22.19) implies (see Figure 11)

z(U(·, t)− w(·, t)) ≥ 2 for t < T, t close to T,

which contradicts (22.20).

Remarks 22.6. (a) Cauchy problem. Let Ω = Rn, u0 ∈ C(Rn) be nonnegative,

bounded and radially symmetric and let the solution u of (22.1) be global. Thenthe boundedness of u is known in each of the following (supercritical) cases:

(i) pS < p < pL, u0 ∈ C1 has compact support and its local minima arebounded away from zero (see [374]);

(ii) pS pJL , u0(x) ≤ U∗(|x|) − c0|x|−|α|, where c0 > 0,

α =12

[−(n− 2) +

√β2 − 4(p− 1)cp−1

p

],

β = n− 2− 4/(p− 1) and cp, U∗ are defined in (3.9) (see [378]).

On the other hand, if p ≥ pJL it was shown in [428] that there exists a continu-ous radial function u0 satisfying 0 < u0(x) ≤ U∗(|x|) such that the correspondingsolution u is global and unbounded. See also Section 29 for more precise informa-tion on the asymptotic behavior of such solutions.

0 R

U∗

w(·, T )

U(·, t)

Figure 11: Graphs of U∗, w(·, T ), U(·, t) if t is close to T .

(b) Inhomogeneous boundary conditions. The result in Theorem 22.4 issensitive to the boundary conditions. Indeed consider problem (22.1) in Ω = B1

with the boundary conditions replaced by u = a > 0 on ∂Ω × (0,∞). Note thatthis is equivalent to problem (14.1) with

f(v) := (v + a)p (resp., f(w) := λ(w + a)p, λ = a1−p)

via the transformation v = u − a (resp., w = a−1u − 1). If pS pJL and a = cp, where cp

is given by (3.9), then there exist unbounded global solutions [316]. More precisely,any initial data u0 ∈ L∞(Ω) satisfying 0 ≤ u0(x) ≤ u∗(x) = U∗(|x|) gives rise toan unbounded global classical solution, which stabilizes to u∗ as t→∞. The rateof approach has been studied in [164].

Remark 22.7. Eventual radial monotonicity of global radial solutions.The following property was shown in [395] (actually for more general nonlinear-ities). Let p > 1, Ω = BR, and assume that u ≥ 0 is a radial, global classical

solution of (22.1) (not necessarily bounded). Then there exists t0 > 0 such that ubecomes radial nonincreasing for t ≥ t0.

Remark 22.8. Exponential nonlinearity. Consider the problemut −∆u = λeu, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (22.21)

where Ω ⊂ Rn, λ > 0 and u0 ∈ L∞(Ω). Embedding theorems, intersection proper-

ties of stationary solutions of (22.21) for Ω = Rn (see [510]), bifurcation diagrams

for stationary solutions of (22.21) for Ω being a ball (see Remark 6.10(ii)), and sev-eral results for time dependent solutions of (22.21) indicate that the cases n ≤ 2,3 ≤ n ≤ 9 and n ≥ 10 correspond to the cases p < pS, pS pJL

for problem (22.1), respectively. In fact, many (but not all) proofs in this chaptercan be adapted to the case of exponential nonlinearity. Unfortunately, similarly asin the case of the power nonlinearity, a lot of basic questions for (22.21) remainopen. For example, a priori bounds (22.2) are known if n = 1 (see [440]) but notfor n = 2. On the other hand, the boundedness of global solutions of (22.21) withgeneral bounded Ω is true if n ≤ 2 (see [186], [109]) and the boundedness of radialglobal solutions in a ball for 3 ≤ n ≤ 9 is known as well (see [198]). We refer to[188] for a survey on problem (22.21).

22.3. Global unbounded solutions in the critical case

The following result due to [232] shows that the situation in the Sobolev criticalcase is very different from both the subcritical and the supercritical cases.

To formulate it, we introduce the notion of threshold solution. Let ϕ ∈ L∞(Ω) bea fixed nonnegative function, ϕ ≡ 0, α > 0, and set u0 = αϕ. If α is small enough,then the solution u = u(t; αϕ) of (22.1) exists globally. (Moreover u(t) → 0 inL∞(Ω), as t→∞.) This follows from Theorem 19.2. We may thus define

α∗ = α∗(ϕ) := supα > 0 : Tmax(αϕ) = ∞. (22.22)

Note that α∗ ∈ (0,∞) due to Remark 17.7(v). The function u∗ = u(t; α∗ϕ) is calledthe threshold solution (associated with ϕ), due to the fact that u∗ lies on theborderline between blow-up and global existence. Further properties of thresholdand non-threshold solutions will be studied in Sections 27 and 28.

Theorem 22.9. Consider problem (22.1) with p = pS and Ω = BR. Let u0 = α∗ϕ,where ϕ(x) = Φ(|x|), with 0 ≤ Φ ∈ L∞(0, R), Φ nonincreasing, and α∗ defined by(22.22). Then the solution u∗ is global and unbounded. More precisely,

limt→∞ ‖u

∗(t)‖q = ∞ for any q > pS + 1,

lim inft→∞ ‖u∗(t)‖pS+1 < ∞.

(22.23)

Proof. First assume that u∗ blows up in finite time T . Let αk α∗, α1 > 0 andlet vk, k = 1, 2, . . . , denote the (global) solution with the initial data αkϕ. Thesolutions u∗, vk are radial and radially decreasing, u∗(x, t) = U∗(|x|, t), vk(x, t) =Vk(|x|, t). Let t1 ∈ (0, T ) be fixed. Since V1 is positive on Q1 := [0, R/2]×[t1, T +1],there exists c1 > 0 such that Vk ≥ V1 > c1 on Q1 for any k. In addition, U∗ ≥ Vk on[0, R/2]× [t1, T ). The functions U∗(·, t1) and Vk(·, t1), k = 1, 2, . . . , are uniformlybounded in C1([0, R]). In particular, there exists c2 > 0 such that Vk(·, t1) ≤U∗(·, t1) < c2.

Let UM be the unique positive solution of (9.2) satisfying UM (0) = M , seeTheorem 9.1. Since UM (R/2) → 0 as M → ∞, there exists M1 > 0 such thatUM (R/2) < c1 for all M ≥ M1. Enlarging M1 if necessary we may also assume thatthe function M → UM (R/2) is decreasing for M ≥ M1, and that M1 ≥ M0(R/2),where the function M0 is defined in Remark 9.3. Finally, since U ′

M (r) → −∞ asM → ∞ uniformly on r : UM (r) ∈ [c1, c2], we may assume that UM intersectsany of the functions Vk(·, t1), k = 1, 2, . . . , exactly once in [0, R/2], for all M ≥ M1,see Figure 12.

0 R/2

U∗(·, t1)

Vk(·, t1)

UM

c1

c2

Figure 12: Graphs of U∗(·, t1), Vk(·, t1), UM .

Consequently, denoting by z(ψ) the zero number of the function ψ in the interval[0, R/2] (see Appendix F), we have

z(UM − Vk(·, t1)) = 1, k = 1, 2, . . . , M ≥ M1. (22.24)

Fix M2 > M1 (see Figure 13) and let U be the solution of the problem

Ut − Urr −n− 1

rUr = Up, r ∈ (0, R/2), t > 0,

Ur(0, t) = 0, U(R/2, t) = UM1(R/2), t > 0,

U(r, 0) = max(UM2(r), UM1(r)), r ∈ (0, R/2).

0 R/2

Vk(·, t1)

UM2

UM1

c1

Figure 13: Graphs of Vk(·, t1), UM1 , UM2 .

We have Ur ≤ 0 for t > 0 by Proposition 52.17. Moreover, the function U(r, 0)is a subsolution for this problem, hence

Ut ≥ 0, for t > 0 (22.25)

(in fact this follows from a simple modification of the proof of Proposition 52.19).We claim that:

U blows up in a finite time T . (22.26)

Assume for contradiction that U exists globally and let V (r) := limt→∞ U(r, t).First we have V (r) <∞ for 0 < r ≤ R/2 (otherwise we would have

limt→∞ U(r, t) =∞ uniformly on [0, r0) for some r0 > 0,

which would imply finite-time blow-up by an eigenfunction argument — cf. theproof of Theorem 17.1). It follows from (22.25) and Lemma 53.10 that V ∈C2((0, R/2]) is a solution of Vrr + n−1

r Vr + V p = 0 on 0 < r ≤ R/2. Moreover wehave V > max(UM2 , UM1) and Vr ≤ 0 on (0, R/2), and V (R/2) = UM1(R/2). Butsince M1 ≥ M0(R/2), Remark 9.3 implies V (r0) = UM1(r0) for some r0 ∈ (0, R/2):a contradiction. Consequently, (22.26) is true.

Fix β ≥ 1 such that Tβ := T β1−p < 1, set Rβ := β−(p−1)/2R/2 and notice thatW (r, t) := βU(rβ(p−1)/2, tβp−1) is a solution of the problem

Wt −Wrr −n− 1

rWr = W p, r ∈ (0, Rβ), t > 0,

Wr(0, t) = 0, W (Rβ , t) = UβM1(Rβ), t > 0,

W (r, 0) = max(UβM2(r), UβM1(r)), r ∈ (0, Rβ),

which blows up at time Tβ < 1. Since U∗ blows up at time T and is decreasingin r, and since Vk(0, t) → U∗(0, t) as k → ∞ for any t < T , there exist k andt0 ∈ (t1, T ) such that Vk(0, t0) > UβM2(0) = βM2 > UβM1(0). Notice also that

Vk(R/2, t) > c1 > UβM1(R/2) > UβM2(R/2) for all t ∈ [t1, T + 1].

Now (22.24) and the monotonicity of the zero number (see Theorem 52.28) implyVk(·, t) > UβMi on [0, R/2] for all t ∈ [t0, T + 1] and i = 1, 2, hence Vk(·, t0) >W (·, 0) on [0, Rβ ]. Since Vk(Rβ , t) > UβM1(Rβ) = W (Rβ , t) for t ∈ [t0, T + 1], wehave Vk(·, t + t0) > W (·, t) whenever t > 0, t + t0 ≤ T + 1. However W blows upat Tβ < 1 which yields a contradiction. Consequently, u∗ is global.

Next assume that lim inf t→∞ ‖u∗(t)‖q < ∞ for some q > pS + 1. Then thereexist C > 0 and tk → ∞ such that ‖u∗(tk)‖q < C. Fix γ ∈ (1/2, 1). Sinceq > n(pS − 1)/2, Theorem 51.25, Remark 51.26(vi) and Example 51.27 (withz = q and α = 1) show the existence of δ > 0 such that the sequence u∗(tk + δ)is bounded in W 2γ,q∩W 1,q

0 (Ω), hence relatively compact in X := H10∩Lq(Ω). Next

Example 51.28 and Proposition 53.6 guarantee that a subsequence of u∗(tk + δ)converges in X to an equilibrium v. The maximum principle implies v ≥ 0. Assumev = 0. Then α∗ϕ belongs to the domain of attraction of the zero solution (which isan open set) hence the same is true for αϕ with some α > α∗. But this contradictsthe definition of α∗. Consequently, v > 0. However, this contradicts Corollary 5.2.

Finally assume limt→∞ ‖u∗(t)‖pS+1 = ∞. Then estimate (17.9) shows that theL2-norm of u∗(t) has to blow up in finite time which is absurd. In fact, Theo-rem 17.6 also shows that the energy of u∗(t) remains bounded and the proof ofTheorem 22.1 guarantees that the norm of u∗ in L4((t, t + 1), H1(Ω)) is boundeduniformly with respect to t ≥ t0 > 0.

Remarks 22.10. (i) Grow-up rates in a ball. The assertion in Theorem 22.9remains true even if the function Φ is not monotone (cf. the proof of Theorem 28.7

below). In addition, if R = 1, then all such global unbounded radial positivesolutions exhibit the following asymptotic behavior as t→∞ (see [223]):

log ‖u∗(·, t)‖∞ =π2

4t(1 + o(1)) if n = 3,

log ‖u∗(·, t)‖∞ = 2√

t(1 + o(1)) if n = 4,

‖u∗(·, t)‖∞ = γ0t(n−2)/2(n−4)(1 + o(1)) if n ≥ 5,

where the constant γ0 > 0 depends only on the spatial dimension n.(ii) Grow-up rates for the Cauchy problem. Let Ω = R

n, p = pS , γ >2/(p− 1), Φ : [0,∞) → (0,∞) satisfy Φ(r) ∼ Cr−γ for r large and considerinitial data u0(x) = α∗Φ(|x|), where α∗ has the same meaning as in Theorem 22.9.If n = 3, then formal matched asymptotics expansions ([306]) suggest that fort large, ‖u(t)‖∞ behaves like t(γ−1)/2 or t1/2 provided γ ∈ (1/2, 2) or γ > 2,respectively. On the other hand, the same arguments indicate that this solutionremains bounded if n > 3.

(iii) Nonuniformity of the smoothing time in the critical Lq-space. Letu∗ be the global unbounded solution from Theorem 22.9. Fix C1 > 0 and tk →∞such that ‖u∗(tk)‖pS+1 < C1. Since pS + 1 = qc = n(pS − 1)/2, Remark 15.4(i)guarantees that problem (22.1) is well-posed in LpS+1(Ω) and, in particular, thereexist C2 > 0 and Tk > 0 such that

‖u∗(tk + t)‖∞ ≤ C2‖u∗(tk)‖pS+1t−α ≤ C1C2t

−α, t ∈ (0, Tk),

where α = (n/2)(pS + 1) = (n − 2)/4, cf. (15.2). Since ‖u∗(tk + t)‖∞ → ∞ forany t ≥ 0, we see that Tk → 0 (in spite of the fact that Tmax(u∗(tk)) = ∞ and‖u∗(tk)‖pS+1 < C1).

22.4. Estimates for nonglobal solutions

The estimates in Theorem 22.1 can be extended to nonglobal solutions in thefollowing way.

Proposition 22.11. Assume Ω bounded, 1 0 and ‖u0‖∞ ≤ K.If u is the solution of (22.1), then

‖u(t)‖∞ ≤ C(δ, K) for all t ∈ [0, Tmax(u0)− δ), (22.27)

(where Tmax(u0)− δ :=∞ if Tmax(u0) = ∞) and

E(u(t)

)→ −∞ as t→ Tmax(u0), whenever Tmax(u0) < ∞. (22.28)

Remark 22.12. Related blow-up rate estimates of the form

‖u(t)‖∞ ≤ M(T − t)−1/(p−1), 0 < t < T := Tmax(u0),

will be proved in Section 23. In some cases the constant M will be known todepend on u0 through a bound on ‖u0‖∞ only (see e.g. Remark 23.9). However,up to now, such a priori estimates are not available under the general assumptionsof Proposition 22.11 (one has to assume either Ω convex, or u ≥ 0 and a strongerrestriction on p)

Proof of Proposition 22.11. If Tmax(u0) = ∞, then estimate (22.27) followsfrom Theorem 22.1.

Assume Tmax(u0) < ∞, and set T := Tmax(u0) − δ. As in the proof of Theo-rem 22.1 we may assume that ‖u0‖1,2 ≤ C, hence E(u(t)) ≤ C for t ≥ 0. Denotingψ(t) = ‖u(t)‖22 we have (cf. (17.9))

12ψ′(t) ≥ −2E(u(t)) + c1ψ

(p+1)/2(t),

where c1 = c1(p, Ω) > 0. Set M := ((p−1)c1δ/2)−2/(p−1)/δ and assume E(u(t0)) ≤−M for some t0 ∈ [0, T ]. Then ψ′(t) ≥ 4M for t ≥ t0, hence ψ(t0 + δ/2) ≥ 2δM .Since ψ′ ≥ 2c1ψ

(p+1)/2 and the solution of the problem

y(0) = 2δM, y′ = 2c1y(p+1)/2

blows up at t < δ/2, ψ cannot exist on the whole interval [t0 + δ/2, t0 + δ) whichyields a contradiction. Consequently, E(u(t)) ≥ −M for all t ∈ [0, T ] and similararguments show ‖u(t)‖2 =

√ψ(t) ≤ M for all t ∈ [0, T ] and suitable M =

M(K, δ). Now (22.27) follows from Remark 22.2(i).Assertion (22.28) follows from Remark 22.2(ii).

Estimates (22.27) and (22.28) can be proved for a fairly general class of su-perlinear subcritical parabolic problems in bounded domains, including problemswith nonlocal nonlinearities (see [440]).

Property (22.28) plays an important role in the proof of complete blow-up (seeRemark 27.8(b) below).

As an easy application of estimate (22.27) we obtain the following importanttheorem concerning the continuity of the existence time.

Theorem 22.13. Assume Ω bounded, 1 < p < pS, and let Tmax(u0) denote themaximal existence time of the solution of (22.1). Then the function

Tmax : L∞(Ω) → (0,∞] : u0 → Tmax(u0)

is continuous.

23. Blow-up rate 177

Proof. If 0 < T < Tmax(u0), then the continuous dependence of solutions of (22.1)on initial data (see (51.28)) guarantees the existence of ε > 0 such that Tmax(v0) >T for any v0 satisfying ‖u0− v0‖∞ < ε. Hence Tmax is lower semicontinuous. Nextassume u0,k → u0 in L∞(Ω) and Tmax(u0,k) > T + δ > 0 for some δ > 0 and all k.Then (22.27) guarantees that the corresponding solutions uk satisfy ‖uk(t)‖∞ ≤ Cfor all t ∈ [0, T ] and k = 1, 2, . . . . Passing to the limit we obtain Tmax(u0) ≥ Tand ‖u(t)‖∞ ≤ C. Consequently, Tmax is upper semicontinuous.

The function Tmax need not be continuous in the supercritical case even in themodel case (22.1) (consider the threshold trajectory u∗ from Theorem 28.7 below:if u∗ blows up in finite time, then Tmax is not continuous at u∗(0) = α∗φ).

23. Blow-up rate

In this section we consider the model problem (22.1) and assume that Tmax(u0) <∞. The solution of the ODE

y′ = yp, t > 0, y(0) = y0 > 0, (23.1)

is given by

y(t) = k(T − t)−1/(p−1), 0 < t < T, where k = (p− 1)−1/(p−1), (23.2)

with T = (p−1)−1y1−p0 . It is natural to ask whether the blow-up rate for (22.1) will

be of the same order. More precisely, do there exist positive constants C1, C2 > 0such that

C1(T − t)−1/(p−1) ≤ ‖u(t)‖∞ ≤ C2(T − t)−1/(p−1), (23.3)

where T := Tmax(u0)? It is not difficult to show that the lower bound in (23.3) isalways satisfied, in fact with the same constant as for the ODE.

Proposition 23.1. Consider problem (22.1) with p > 1. Let u0 ∈ L∞(Ω) andassume that T := Tmax(u0) < ∞. Then

‖u(t)‖∞ ≥ k(T − t)−1/(p−1), 0 < t < T.

Proof. Assume for contradiction that there exists t0 ∈ [0, T ) such that ‖u(t0)‖∞< y(t0), where y is given by (23.2). Therefore ‖u(t0)‖∞ ≤ y(t0−ε) for some ε > 0.Since y′ = yp, we deduce from the comparison principle that ±u(x, t) ≤ y(t−ε) for(x, t) ∈ Ω× (t0, T ). If follows that u is bounded in Ω× (t0, T ), a contradiction.

We present an alternative proof from [219]. It is slightly less simple but theargument may be useful for other problems (see e.g. the proofs of Theorems 44.2(i),44.17(ii) and 46.4(i)).

Alternative proof for Ω bounded and u0 ≥ 0. We may assume

M(t) := maxx∈Ω

u(x, t) > 0

for all t ∈ (0, T ) and pick x0(t) ∈ Ω such that M(t) = u(x0(t), t). For 0 < s < t <T , we have

M(t)−M(s) ≤ u(x0(t), t)− u(x0(t), s) = (t− s)ut

(x0(t), s + θ(t− s)

)(23.4)

and

M(t)−M(s) ≥ u(x0(s), t)− u(x0(s), s) = (t− s)ut

(x0(s), s + θ(t− s)

)for some θ, θ ∈ (0, 1). Since ut is locally bounded in Ω× (0, T ), it follows that thefunction M is locally Lipschitz. In particular, M is a.e. differentiable.4 Dividing(23.4) by t− s, passing to the limit s → t, and using ∆u(x0(t), t) ≤ 0, we obtain

M ′(t) ≤ ut(x0(t), t) ≤ up(x0(t), t) = Mp(t), a.e. in (0, T ).

Integrating between t and s ∈ (t, T ) we get M1−p(t) ≤ M1−p(s) + (p − 1)(s − t)and the conclusion follows by letting s → T and using lims→T M(s) =∞.

Remarks 23.2. (i) Radial case. In the case when Ω = BR and u ≥ 0 is radialdecreasing in r, then the above proof is just reduced to the obvious observationthat x0(t) = 0 and M ′(t) = ut(0, t) ≤ up(0, t) = Mp(t).

(ii) Alternative proof. By simple arguments based on the variation-of-con-stants formula, one obtains still another proof (cf. [530]) of the lower bound in(23.3) (without the sharp constant). Indeed, by (15.5), we have

‖u(s)‖∞ ≤ ‖u(t)‖∞ +∫ s

t

‖u(τ)‖p∞ dτ, 0 < t < s < T

and, by choosing s = minτ ∈ (t, T ) : ‖u(τ)‖∞ = 2‖u(t)‖∞, we obtain ‖u(t)‖∞ =‖u(s)‖∞−‖u(t)‖∞ ≤ 2p(T−t)‖u(t)‖p

∞, hence the lower bound in (23.3). For similarestimates concerning Lq-norms (also based on the variation-of-constants formula),see Remark 16.2(iii).

(iii) Estimation of the blow-up time. An upper estimate of the blow-uptime was given in Remark 17.2(i). Proposition 23.1 provides the lower estimate

Tmax(u0) ≥1

p− 1‖u0‖1−p

∞ .

4Alternatively, one could avoid employing this fact and use an argument involving the deriv-ative of M in the sense of distributions.

The upper blow-up rate estimate

‖u(t)‖∞ ≤M(T − t)−1/(p−1), 0 ≤ t < T (23.5)

(for some constant M > 0 possibly depending on u) is much less trivial andneed not be always true. It was first obtained in [531], [534] for special classesof solutions. Estimate (23.5) is sometimes referred to as type I blow-up, whereasblow-up is said to be of type II if (23.5) fails (cf. [355]). In this section we provethis upper estimate in three cases:

(i) for all p > 1 when u ≥ 0 is increasing in time, with Ω bounded — cf. Theo-rem 23.5. This result is due to [219] if Ω is convex; similar ideas were usedbefore in [502] to estimate blow-up times.

(ii) for 1 < p < pS when u ≥ 0 and Ω = Rn (cf. Theorem 23.7, a result due to

[245]);(iii) for pS ≤ p < pJL when Ω = BR and u ≥ 0 is radial nonincreasing (cf. The-

orem 23.10, a result due to [355], see also [206] for a related result in thecase p = pS).

On the contrary, type II blow-up may occur if Ω = Rn, n ≥ 11 and p > pJL:

There exist radial nonincreasing solutions u ≥ 0 such that (23.5) fails (see [277],[278] and [377]). The proof of this important result is quite long and delicate andwill not be given here. Formal arguments indicate that this upper estimate shouldalso fail for some radial (sign-changing) solutions if Ω = R

n, 3 ≤ n ≤ 6 and p = pS

(see [206]). If p ≥ pS , nothing seems to be known for solutions which are neitherradial nor increasing in time.

Remarks 23.3. (a) Extensions. The result of case (i) above remains true forΩ = R

n if we assume in addition that u0 is radial nonincreasing (see [358]). Theresult of case (ii) is true also for Ω bounded convex (see [245]) and without theassumption u ≥ 0 (see [247], [248]). If u ≥ 0 and p < pB, the convexity assumptioncan be removed (see Theorem 26.8 below). As for the result of case (iii), it remainstrue for all radial solutions if pS < p < pJL and for all positive radial solutions ifp = pS (see [355]). In the case Ω = R

n, it is true under an additional assumptionon u0.

(b) Different methods of proof. The three proofs corresponding to cases (i),(ii) and (iii) above are quite different. They are based respectively on the maximumprinciple (applied to a suitable auxiliary function), on similarity variables, rescal-ing and energy, and on rescaling and intersection-comparison. In particular cases,different rescaling (resp., intersection-comparison) arguments were used before in[534] (resp., [229]).

(c) Neumann problem. For problem (22.1) with Neumann instead of Dirichletboundary conditions, results on (type I) blow-up rate can be found in [219], [375],for example.

Remark 23.4. Refined blow-up rate estimates. Assume p < pS and u ≥ 0.When Ω = R

n or Ω is a bounded convex domain, the refined asymptotic behaviorlimt→T (T − t)1/(p−1)‖u(t)‖∞ = k was proved in [367] (see also [368] for a higherorder asymptotic expansion). On the other hand, estimates similar to (23.5), butwith M independent of u, will be studied in Section 26 on universal bounds.

Theorem 23.5. Consider problem (22.1) with p > 1, Ω bounded and 0 ≤ u0 ∈L∞(Ω). Assume that u is nondecreasing in time and nonstationary. Then T :=Tmax(u0) <∞ and blow-up is of type I, i.e. (23.5) is true.

Remark 23.6. The assumption ut ≥ 0 is guaranteed if, for instance, 0 ≤ u0 ∈C0 ∩ C2(Ω) and ∆u0 + up

0 ≥ 0 (see Proposition 52.19, and also Proposition 52.20for weaker regularity conditions on u0).

Proof of Theorem 23.5. It is a modification of the corresponding proof in [219].The idea is to apply the maximum principle to the auxiliary function J definedin (23.6) below. By Example 51.10, we have ut ∈ C2,1(Ω×(0, T )). Since v := ut ≥ 0is a nontrivial solution of vt−∆v = f ′(u)v in QT vanishing on ST , it follows fromthe Hopf maximum principle (cf. Proposition 52.7) that ut > 0 in QT and ∂νut < 0on ST . Choosing η ∈ (0, T ) we can thus find δ > 0 such that ut(x, η) ≥ δup(x, η)for all x ∈ Ω. Set

J := ut − δf, where f = f(u) := up, (23.6)

and note that J ∈ C2,1(QT ) ∩ C(Ω× (0, T )) (due to u > 0 in QT ). We compute

Jt −∆J = f ′ut − δf ′f + δf ′′|∇u|2,

henceJt −∆J − f ′J = δf ′′|∇u|2 ≥ 0 in Qη := Ω× (η, T ).

Since J ≥ 0 on the parabolic boundary of Qη, it follows from the maximumprinciple (cf. Proposition 52.4) that J ≥ 0 in Qη. Consequently, ut ≥ δup in Qη.For each x ∈ Ω, by integrating this inequality between t and s ∈ (t, T ), and thenletting s → T , we obtain

u1−p(x, t) ≥ (p− 1)δ(T − t), η < t < T.

This gives T < ∞ and (23.5).

Theorem 23.7. Consider problem (22.1) with Ω = Rn, 1 < p < pS, 0 ≤ u0 ∈

L∞(Rn), and assume that T := Tmax(u0) < ∞. Then blow-up is of type I, i.e.(23.5) is true.

In view of the proof of Theorem 23.7 we introduce the notion of backwardsimilarity variables (cf. [244], [228]). This is a fundamental tool in the study

of the asymptotic behavior of blow-up solutions to problem (22.1), and it will beused again in Section 25. Namely, let 0 < T < ∞ and let u be a solution of (22.1)with Ω = R

n, such that u exists on Rn × (0, T ). For each fixed a ∈ R

n, we set

y :=x− a√T − t

, s := − log(T − t), (23.7)

and we define the rescaled function

w(y, s) = wa(y, s) := e−βsu(a + e−s/2y, T − e−s), β :=1

p− 1(23.8)

(in other words, w(y, s) = (T − t)βu(x, t)). Let s0 := − log T . Then w is a globalsolution of

ws −∆w +12y · ∇w = |w|p−1w − βw, y ∈ R

n, s ∈ (s0,∞), (23.9)

withw(y, s0) = T βu0(a + y

√T ), y ∈ R

n. (23.10)

Observe that (23.5) is equivalent to the uniform estimate |wa(y, s)| ≤ M , whereM does not depend on y, s. To prove (23.5) we shall use this fact and the rescalingarguments from the proof of Theorem 22.1 for nonnegative solutions.

Note that equation (23.9) can be rewritten as

ρws −∇ · (ρ∇w) = ρ|w|p−1w − βρw, (23.11)

where the Gaussian weight ρ is defined by

ρ(y) := e−|y|2/4.

An important property of the rescaled equation (23.9) is the existence of a weight-ed energy functional, defined by

E(w) :=∫

Rn

(12|∇w|2 +

β

2w2 − 1

p + 1|w|p+1

)ρ dy. (23.12)

We shall first establish some auxiliary results involving this energy (these resultswill also be used in Section 25).

Proposition 23.8. Let p > 1 and let w be a global solution of (23.11) withw(·, s0) ∈ BC1(Rn). Then, for all s > s0, we have

12

d

ds

∫Rn

w2ρ dy = −2E(w(s)

)+

p− 1p + 1

∫Rn

|w|p+1ρ dy, (23.13)

d

dsE(w(s)

)= −

∫Rn

w2sρ dy, (23.14)

E(w(s)

)≥ 0, (23.15)

and ∫Rn

w2ρ dy ≤ C(n, p)[E(w(s0)

)]2/(p+1). (23.16)

Moreover, ∫ ∞

s0

∫Rn

w2sρ dy ds ≤ E

(w(s0)

)(23.17)

anda → E

(wa(s0)

)is smooth and bounded. (23.18)

Proof. Problems (18.1) and (23.9)-(23.10) are equivalent via the transformation(23.7)–(23.8). Let 0 < t1 < t2 < T . By Proposition 48.7 and a simple use ofthe variation-of-constants formula, we see that u,∇u ∈ L∞(Rn × (0, t2)) (seealso (51.29) in Remark 51.11). On the other hand, applying interior parabolicLq- and Schauder estimates, we obtain that D2u, ut ∈ L∞(Rn × (t1, t2)). Nextapplying Remark 48.3(i) and, again, interior Schauder estimates, we get ∇u ∈C2,1(Rn × (0, T )) and ∇ut ∈ L∞(Rn × (t1, t2)). Consequently, given s2 ∈ (s0,∞),the rescaled function w = w(s) satisfies

supRn×(s0,s2)

(|w| + |∇w|) < ∞ (23.19)

and

supRn×(s1,s2)

(|D2w|+ (1 + |y|)−1(|ws|+ |∇ws|)

)< ∞, s0 < s1 < s2. (23.20)

We shall write shortly∫

f instead of∫

Rn f(y) dy. We compute

12

d

ds

∫w2ρ =

∫wwsρ =

∫w[∇ · (ρ∇w) + ρ|w|p−1w − βρw

],

and12

d

ds

∫|∇w|2ρ =

∫ρ(∇ws · ∇w),

for s > s0. Note that the differentiability of the integrals is guaranteed by (23.19),(23.20), and the exponential decay of ρ. By using integration by parts, we deducethat

12

d

ds

∫w2ρ =

∫ [−|∇w|2 − βw2 + |w|p+1

]ρ = −2E(w) +

p− 1p + 1

∫|w|p+1ρ


i.e., (23.13), and12

d

ds

∫|∇w|2ρ = −

∫ws(∇ · ρ∇w).

This procedure can be easily justified by using again (23.19), (23.20), and theexponential decay of ρ: It suffices to integrate by parts on BR and then let R →∞.On the other hand, we have

d

ds

∫ (β

2w2 − 1

p + 1|w|p+1

)ρ =

∫(βw − |w|p−1w)wsρ.

Summing the last two identities and using equation (23.11), we obtain (23.14).Denote ψ(s) :=

∫w2(s)ρ. Then (23.13) and Jensen’s inequality imply

12

dψ

ds≥ −2E

(w(s)

)+C(n, p)ψ(p+1)/2(s).

Since E(w(s)

)is nonincreasing due to (23.14), this guarantees (23.15) and (23.16)

(otherwise ψ has to blow up in finite time). Next, (23.15) and (23.14) imply (23.17).Finally, to check (23.18), we note that∫

w2a(s0)ρ dy = T 2β

∫Rn

u20(x)ρ

(x− a√T

) dx

T n/2≤ T 2β(4π)n/2 sup u2

0,

which shows the smoothness and boundedness of the second term appearing inthe definition of E

(wa(s0)

), see (23.12). The proof for the remaining terms is

similar.

Now we are ready to repeat the idea of the proof of Theorem 22.1 for nonnegativesolutions.

Proof of Theorem 23.7. By a time shift we may assume u0 ∈ BC1(Rn), see(51.28). Assume, on the contrary, that there exist tk such that

Mk := supRn×[0,tk]

(T − t)βu(x, t) = supRn

(T − tk)βu(x, tk) →∞.

We may assume tk ≥ t for some t > 0. Choose xk ∈ Rn such that

(T − tk)βu(xk, tk) ≥ Mk/2.

Rewriting u in similarity variables around a = xk (cf. (23.7)–(23.8)), we denotewk := wxk

, sk := − log(T − tk). Then sk − s0 ≥ δ2 for some δ > 0, 0 ≤ wk(y, s) ≤Mk for s ≤ sk and wk(0, sk) ∈ [Mk/2, Mk]. Denote

vk(z, τ) :=1

Mkwk(νkz, ν2

kτ + sk), νk := M−(p−1)/2k .

Then 0 ≤ vk(z, τ) ≤ 1 for (z, τ) ∈ Qk := Rn× (−(δ/νk)2, 0], vk(0, 0) ∈ [1/2, 1] and

∂τvk −∆vk = vpk − ν2

k

(12z · ∇vk + βvk

)in Qk.

Since Q(r) := (z, τ) : |z| < r, −r2 < τ ≤ 0 ⊂ Qk for k large enough, uniformparabolic Lp-estimates used for the operators Akv := −∆v + 1

2ν2kz · ∇v (see Ap-

pendix B) imply the boundedness of vk in Cα. Consequently, we may pass to thelimit to get a solution v of the problem

vτ −∆v = vp in Rn × (−∞, 0),

satisfying 0 ≤ v ≤ 1 and v(0, 0) ≥ 1/2. Finally, setting σ := −n+2+4/(p−1) > 0and using (23.17) and (23.18) we obtain

∫∫Q(δ/νk)

|∂τvk|2 dz dτ = νσk

∫ sk

sk−δ2

∫|y|<δ

|∂swk|2 dy ds

≤ νσk C(δ)

∫ ∞

s0

∫Rn

|∂swk|2ρ(y) dy ds → 0 as k →∞,

hence vτ ≡ 0 and we get a contradiction with Theorem 8.1.

Remark 23.9. A priori estimate of the blow-up rate. A simple modificationof the proof shows that in Theorem 23.7, the constant M in (23.5) depends on u0

through a bound on ‖u0‖∞ only. The same property is true in the case of boundedconvex domains (this follows from the arguments in [245], see also [368]).

Theorem 23.10. Consider problem (22.1) with pS ≤ p < pJL and Ω = BR. Letu0 ∈ L∞(Ω), u0 ≥ 0, be radial nonincreasing. If T := Tmax(u0) < ∞, then blow-upis of type I, i.e. (23.5) is true.

As a preparation to the proof, we first derive the following result, valid forall p ≥ pS and of independent interest. It shows that in case of type II blow-up (i.e. if (23.5) is violated) or of unbounded global solutions, suitably rescaledsolutions should converge along some sequence to the positive radial steady stateU1, solution of

U ′′ +n− 1

rU ′ + Up = 0, r ∈ (0,∞),

U(0) = 1, U ′(0) = 0

⎫⎬⎭ (23.21)

(which is known to be unique, cf. Theorem 9.1).

Proposition 23.11. Consider problem (22.1) with p ≥ pS and Ω = BR. Letu0 ∈ L∞(Ω), u0 ≥ 0, be radial nonincreasing, and set T := Tmax(u0). Assume thateither

T < ∞ and lim supt→T

(T − t)1/(p−1)‖u(t)‖∞ = ∞, (23.22)

orT = ∞ and lim sup

t→T‖u(t)‖∞ = ∞. (23.23)

Then there exists a sequence tj → T such that

1m(tj)

u

(r

mp−12 (tj)

, tj

)→ U1(r), j →∞, (23.24)

uniformly for bounded r ≥ 0, where m(t) := u(0, t) and U1 = U1(r) is the uniquesolution of (23.21).

In the case T < ∞, Proposition 23.11 was actually established in [355] forgeneral radial solutions without assuming u ≥ 0 nor ur ≤ 0 (replacing m(t) by‖u(t)‖∞ and U1 by ±U1). Here in the radial decreasing case, we give a simplerproof, which is due to [125] (and which applies to T ≤ ∞). Theorem 23.10 willthen be deduced as a consequence of intersection-comparison arguments involvingU1 and the singular steady state U∗.

In the proof of Proposition 23.11, we shall need the following general mono-tonicity property of unbounded, positive radial nonincreasing solutions, valid forall p > 1 (see [395], [229]).

Lemma 23.12. Consider problem (22.1) with p > 1 and Ω = BR. Let u0 ∈L∞(Ω), u0 ≥ 0, be radial nonincreasing, and set T := Tmax(u0). Assume thateither T < ∞ or (23.23) holds. Denote by N(t) := z[0,R](ut(·, t)) the zero numberof the function ut(·, t) in the interval [0, R] (see Appendix F). Then there existst0 ∈ (0, T ) such that

ut(0, t) > 0 and N(t) = Const., t0 < t < T. (23.25)

Proof. We note that the function ut is a radial classical solution of

Vt −∆V = pup−1V, x ∈ BR, 0 < t < T,

V = 0, x ∈ ∂BR, 0 < t < T.

(23.26)

By Theorem 52.28, N(t) is finite and nonincreasing, hence constant on (t0, T ) forsome t0 ∈ (0, T ). Moreover, by the symmetry of the solution, if ut(0, t) = 0, thenut(·, t) has a degenerate zero at r = 0, so that the function N drops at time t.Consequently, ut(0, t) does not change sign on (t0, T ). Since lim supt→T u(0, t) =∞, the claim follows.

Proof of Proposition 23.11. By a time shift, we may assume that t0 = 0 inLemma 23.12. Our assumptions imply the existence of a sequence tj → T suchthat

ut(0, tj)up(0, tj)

→ 0 (23.27)

(otherwise we would have ut(0, t) ≥ cup(0, t) as t → T for some c > 0, due to(23.25); but this contradicts each of (23.22) and (23.23)). Set Mj = u(0, tj). Bycomparing with the solution of the ODE ψ′ = ψp, ψ(tj) = Mj , we easily obtainthe existence of s∗ = s∗(p) > 0 such that

tj := tj + s∗M1−pj < T and u(0, t) ≤ 2Mj, tj ≤ t ≤ tj. (23.28)

Let λj = M−(p−1)/2j and define the rescaled solutions

vj(y, s) =1

Mju(λjy, tj + λ2

js), (y, s) ∈ Dj := BRλ−1j×(−tjλ

−2j , (T − tj)λ−2

j

).

Then∂svj −∆vj = vp

j , (y, s) ∈ Dj

and, by (23.28) and (23.25), we have 0 ≤ vj ≤ 2 in BRλ−1j×(−tjλ

−2j , s∗). Moreover,

vj(0, 0) = 1 and ∂svj(0, 0) → 0, due to (23.27). Let D := Rn × (−∞, s∗). By

interior parabolic estimates, it follows that (some subsequence of) vj converges inC2+α,1+α/2(D) to a radial, nonnegative solution of

∂sv −∆v = vp, (y, s) ∈ D,

such that v(0, 0) = 1 and ∂sv(0, 0) = 0. By using equation (23.26), we see alsothat

∂svj → ∂sv in C1,0(D). (23.29)

We shall now show that ∂sv(·, 0) ≡ 0. Suppose not. Then there exist A > 0 andε ∈ (0, s∗) such that

∂sv(A, s) = 0, |s| ≤ ε. (23.30)

Since ∂sv(·, 0) has a degenerate zero at r = 0, it follows from Theorem 52.28 thatthe zero number of ∂sv on [0, A] drops at s = 0. Namely, we can fix −ε < s1 <0 < s2 < ε such that ∂sv(·, si) has only simple zeroes on [0, A] and such that

z[0,A]

(∂sv(·, s1)

)≥ z[0,A]

(∂sv(·, s2)

)+1.

Owing to (23.29), we deduce that for j large enough,

z[0,A]

(∂svj(·, s1)

)≥ z[0,A]

(∂svj(·, s2)

)+1,

hencez[0,Aλj ]

(ut(·, tj + λ2

js1))≥ z[0,Aλj]

(ut(·, tj + λ2

js2))+1. (23.31)

Since, on the other hand, (23.30) implies ut(Aλj , tj + λ2js) = 0 for |s| ≤ ε, Re-

mark 52.29(ii) implies

z[Aλj ,R]

(ut(·, tj + λ2

js1))≥ z[Aλj ,R]

(ut(·, tj + λ2

js2)). (23.32)

By (23.31) and (23.32), we deduce that N(tj + λ2js1) ≥ N(tj + λ2

js2) + 1, whichcontradicts (23.25). It follows that vs(·, 0) ≡ 0, hence v(·, 0) ≡ U1 due to v(0, 0) =1, and the proposition follows.

Proof of Theorem 23.10. Assume that (23.5) is false and let the sequencetj → T < ∞ be given by Proposition 23.11. We treat the supercritical and criticalcases separately.

Case 1: pS < p < pJL. By Theorem 52.28, there exists an integer K such that

z[0,R](u(·, tj)− U∗) ≤ K, j = 1, 2, . . . . (23.33)

On the other hand, by Theorem 9.1, U1 and U∗ intersect infinitely many times.Moreover, these intersections are transversal by local uniqueness for the ODEU ′′ + n−1

r U ′ + Up = 0. Pick A > 0 such that

z[0,A](U1 − U∗) ≥ K + 1. (23.34)

Also it is clear that

z[0,R]

(u(r, tj)− U∗(r)

)= z[

0,R mp−12 (tj)

] ( 1m(tj)

u

(r

mp−12 (tj)

, tj

)− 1

m(tj)U∗

(r

mp−12 (tj)

))

= z[0,R m

p−12 (tj)

] ( 1m(tj)

u

(r

mp−12 (tj)

, tj

)− U∗(r)

).

By (23.34) and (23.24), it follows that z[0,R]

(u(·, tj) − U∗

)≥ K + 1 for j large: a

contradiction with (23.33).Case 2: p = pS . Fix t0 ∈ (0, T ) and take c2 > c1 > 0 such that

u(r, t) ≥ (e−tAu0)(r) ≥ c1, 0 ≤ r ≤ R/2, t0 ≤ t < T

andu(r, t0) ≤ c2, 0 ≤ r ≤ R.

Let UM be the unique positive solution of (9.2) satisfying UM (0) = M , seeTheorem 9.1. Since UM (R/2) → 0 as M → ∞ and U ′

M (r) → −∞ as M → ∞

uniformly on r : UM (r) ∈ [c1, c2], there exists M0 > c2 such that, for all M ≥M0, u(R/2, t) > UM (R/2) for t ∈ [t0, T ) and z[0,R/2](u(·, t0) − UM ) = 1. By thenonincreasing property of the zero-number (see Theorem 52.28), we deduce that

z[0,R/2]

(u(·, t)− UM

)≤ 1, t0 ≤ t < T, M ≥ M0.

Since limt→T u(0, t) = ∞, for each M ≥ M0, there exists a first τ(M) ∈ (t0, T )such that u(0, τ(M)) = UM (0) = M . By symmetry of the solutions, u(·, τ(M)) hasa double zero at the origin. Therefore, by Theorem 52.28(iii), z[0,R/2](u(·, t)−UM )must drop at t = τ(M), hence

u(r, t) > UM (r), 0 ≤ r ≤ R/2, τ(M) < t < T. (23.35)

Now, for large j, (23.24) implies

u(0, tj) >m(tj)

2U1(0) = Um(tj)/2(0).

Consequently, tj > τ(m(tj)/2). Using (23.35) and (9.4), it follows that

u(r, tj) > Um(tj)/2(r) = m(tj)U1/2(mp−12 (tj)r), 0 ≤ r ≤ R/2, tj ≤ t < T.

Therefore,

1m(tj)

u

(ρ

mp−12 (tj)

, tj

)> U1/2(ρ), 0 ≤ ρ ≤ (R/2)m

p−12 (tj), tj ≤ t < T.

Using (23.24) again and letting j →∞, we obtain

U1(ρ) ≥ U1/2(ρ), 0 ≤ ρ < ∞,

contradicting Theorem 9.1.

Remark 23.13. By combining Proposition 23.11 for T = ∞ and Case 1 of theproof of Theorem 23.10, we obtain an alternative proof [125] of Theorem 22.4 onboundedness of global solutions in the case pS < p < pJL and ur ≤ 0. Moreover, asa consequence of Remark 22.7, this proof can be used without assuming ur ≤ 0.

Remarks 23.14. (i) Sign-changing solutions. The proof of Theorem 23.7 canbe considered as an analogue to the proof of Theorem 22.1 for nonnegative solu-tions. In [247] the authors prove Theorem 23.7 without the positivity assumptionon u0 and the proof is again an analogue of the (interpolation) proof of Theo-rem 22.1 in the general, sign-changing case. However, the localization of the ar-guments of this interpolation proof is nontrivial: The authors of [247] have to usetwo kinds of localized version of weighted energies,

Eϕ(w) :=12

∫Rn

(|∇(ϕw)|2 +

(βϕ2 − |∇ϕ|2

)w2)ρ dy − 1

p + 1

∫Rn

ϕ2|w|p+1ρ dy,

Eϕ(w) :=12

∫Rn

ϕ2(|∇w|2 + βw2

)ρ dy − 1

p + 1

∫Rn

ϕ2|w|p+1ρ dy,

and the corresponding bounds Eϕ(w) ≥ 0, Eϕ(w) ≤ C and |Eϕ(w) − Eϕ(w)| ≤ C.

(ii) Applications of blow-up rate estimates. The knowledge of the blow-up rate has important consequences in the study of the blow-up behavior. Inparticular (23.5) is the first step in the description of asymptotically self-similarblow-up (see Section 25). On the other hand, it can be used for the proof of theHolder continuity of the maximal existence time Tmax : L∞(Ω) → (0,∞] (see [254],[255] and cf. Theorem 22.13).

We close this section with a simple result which shows that the upper blow-upestimate (23.5) implies a similar estimate for the gradient. This property will beuseful in Section 25.

Proposition 23.15. Consider problem (22.1) with p > 1 and u0 ∈ L∞(Ω). As-sume that T := Tmax(u0) <∞ and that (23.5) is satisfied for some M > 0. Then

‖∇u(t)‖∞ ≤M1(T − t)−1/(p−1)−1/2, T/2 ≤ t < T

for some M1 = M1(M, p, Ω, T ) > 0.

Proof. Fix T/2 ≤ t < T and put s = 2t−T ∈ [0, t). By the variation-of-constantsformula, the gradient estimate in Proposition 48.7 and (23.5), we have

‖∇u(t)‖∞ ≤ ‖∇e−(t−s)Au(s)‖∞ +∫ t

s

‖∇e−(t−τ)A|u|p−1u(τ)‖∞ dτ

≤ C(t− s)−1/2‖u(s)‖∞ + C

∫ t

s

(t− τ)−1/2‖u(τ)‖p∞ dτ

≤ CM(t− s)−1/2(T − s)−1/(p−1) + CMp

∫ t

s

(t− τ)−1/2(T − τ)−p/(p−1) dτ.

Since T − t = t− s = (T − s)/2, we have

‖∇u(t)‖∞

≤ 2−1/(p−1)CM(T − t)−1/(p−1)−1/2 + CMp(T − t)−p/(p−1)

∫ t

s

(t− τ)−1/2 dτ

≤ C[2−1/(p−1)M + 2Mp] (T − t)−1/(p−1)−1/2

and the proposition is proved.

24. Blow-up set and space profile

This and the subsequent section are devoted to the space and space-time descrip-tion of singularities of blowing-up solutions of the model problem (22.1). Assumethat the solution u blows up in finite time T := Tmax(u0) and denote by B(u0) itsblow-up set:

B(u0) := x ∈ Ω : ∃(xk, tk) ∈ Ω× (0, T ) such that tk → T and |u(xk, tk)| → ∞.(24.1)

The following theorem, due to [219], guarantees single-point blow-up for radialdecreasing solutions and provides an upper estimate for the blow-up profile.

Theorem 24.1. Consider problem (22.1) with p > 1 and Ω = BR. Let u0 ∈L∞(Ω), u0 ≥ 0, be radial nonincreasing, and assume T := Tmax(u0) < ∞. ThenB(u0) = 0 and, for any α > 2/(p− 1), there exists Cα > 0 such that

u(x, t) ≤ Cα|x|−α, 0 < |x| < R, 0 < t < T.

Corollary 24.2. Under the assumptions of Theorem 24.1, we have

lim supt→T

‖u(t)‖q < ∞, 1 ≤ q < qc = n(p− 1)/2.

Proof of Theorem 24.1. As in the proof of Theorem 23.5, the idea is to applythe maximum principle to a (different) auxiliary function J , defined in (24.3)below. Note that our assumptions imply

ur(r, t) < 0 for all r ∈ (0, R], t ∈ (0, T ), (24.2)

due to Proposition 52.17. We split the proof in two steps.Step 1. Let γ ∈ (1, p), η ∈ (0, T ) and δ > 0. We will show that there exists ε > 0

such that J(r, t) ≤ 0 in Ω× (η, T ), where

J =J(u) := w + c(r)F (u),

w(r) := rn−1ur(r, t), c(r) := εrn+δ, F (u) := uγ .

(24.3)

Denote f(u) := up, Ω1 := Ω ∩ x : x1 > 0 and notice that v := ux1 satisfiesvt − ∆v = f ′(u)v in Ω1. Since v = 0 for x ∈ ∂Ω1, x1 = 0, and v < 0 forx ∈ ∂Ω1, x1 > 0, the maximum principle implies v < 0 for x ∈ Ω1 and vx1(0, t) =ux1x1(0, t) < 0 for t > 0. Hence urr(0, t) < 0. Since J = rn−1(ur + εr1+δuγ),this inequality and (24.2) imply J(r, η) ≤ 0 for all r provided ε is small enough.We have also J ∈ C2,1((0, R) × (0, T )) ∩ C([0, R] × (0, T )) (due to u > 0), with

24. Blow-up set and space profile 191

J(R, t) ≤ 0 and J(0, t) = 0 for t > 0. Now the claim follows from the maximumprinciple in Proposition 52.4, provided we show

Jt +n− 1

rJr − Jrr − bJ ≤ 0 in (0, R)× (η, T ), (24.4)

where b is bounded on (0, R)× (η, T − τ) for all τ > 0. Summing the identities

Jt = wt + cF ′ut,

n− 1r

Jr =n− 1

r

(wr + cF ′ur + c′F

),

−Jrr = −(wrr + cF ′urr + cF ′′u2

r + 2c′F ′ur + c′′F),

and using the equations

ut − urr −n− 1

rur = f(u), wt +

n− 1r

wr − wrr = f ′(u)w,

and F ′′ ≥ 0 we get

Jt +n− 1

rJr − Jrr ≤ f ′w + cF ′f +

2(n− 1)r

cF ′ur +n− 1

rc′F − 2c′F ′ur − c′′F.

Using w = −cF + J and ur = w/rn−1 = (−cF + J)/rn−1 we see that the RHSof the last inequality can be written in the form bJ − cH , where the functionb = f ′ +2(n−1)cF ′r−n−2c′F ′r1−n is bounded on (0, R)× (η, T − τ) for all τ > 0and

H = Ff ′ − F ′f − 2rn−1

F ′F(c′ − n− 1

rc)

+F

c

(c′′ − n− 1

rc′).

Now H ≥ 0 is equivalent to

(p− γ)up−1 + (n + δ)δr−2 ≥ 2εγ(1 + δ)uγ−1rδ (24.5)

which is obviously satisfied if ε is small enough. Consequently, (24.4) is true.

Step 2. Let t ∈ (η, T ) and α := (2 + δ)/(γ − 1). Notice that J(u) ≤ 0 impliesur/uγ ≤ −εr1+δ. Integrating this inequality we arrive at u(r, t) ≤ Cr−α, whereC := (α/ε)1/(γ−1). This estimate guarantees the assertion.

Under an additional assumption of monotonicity in time, the correspondinglower estimate on the blow-up profile can be established by relatively simple ar-guments (cf. [490]). For more precise information on the blow-up profiles, see Re-mark 25.8 below.

Theorem 24.3. Consider problem (22.1) with p > 1 and Ω = BR or Ω = Rn. Let

u0 ∈ L∞(Ω), u0 ≥ 0, be radial nonincreasing and such that T := Tmax(u0) < ∞.Assume in addition that ut ≥ 0 in QT . Then there holds

u(x, T ) ≥ C|x|−2/(p−1), 0 < |x| < η, (24.6)

for some C = C(p) > 0 and η = η(u0) > 0.

Proof. We assume Ω = BR (the case Ω = Rn can be treated by straightforward

modifications). Since ut ≥ 0 and ur ≤ 0, we have

∂

∂r

(12u2

r +1

p + 1up+1

)= (urr + up)ur =

(ut −

n− 1r

ur

)ur ≤ 0,

hence (12u2

r +1

p + 1up+1

)(r, t) ≤ 1

p + 1up+1(0, t).

Therefore, we get‖ur(t)‖∞ ≤ C1u

(p+1)/2(0, t).

For 0 < t < T , let r0(t) be such that u(r0(t), t) = 12u(0, t). Note that, in view

of (24.2), the implicit function theorem guarantees that r0(t) is unique and is acontinuous function of t. We may assume that 0 is the only blow-up point, sinceotherwise the result is trivial. Using u(0, t) = ‖u(t)‖∞ →∞ as t→ T , this impliesr0(t) → 0 as t→ T . Now we have

−ur ≤ C2u(p+1)/2, 0 ≤ r ≤ r0(t).

Integrating, we get

u−(p−1)/2(r0(t), t) ≤ u−(p−1)/2(0, t) + C3r0(t)

= 2−(p−1)/2u−(p−1)/2(r0(t), t) + C3r0(t)

hence u(r0(t), t) ≥ C4(r0(t))−2/(p−1). Using ut ≥ 0, it follows that

u(r0(t), T ) ≥ C4(r0(t))−2/(p−1), 0 < t < T.

Since r0 is continuous and r0(t)→ 0 as t→ T , we deduce that the range r0((0, T ))contains an interval of the form (0, η) and the conclusion follows.

Remark 24.4. The lower bound (24.6) has been proved in [232] (see also [229])for radial solutions without assuming ut ≥ 0 nor ur ≤ 0, but under the conditionp ≤ pS . The method therein is different, based on intersection-comparison withstationary solutions.

We get back to the question of the blow-up set. In the case of Rn, the following

result, due to [246], gives a necessary condition, involving the weighted energy, fora given point to be a blow-up point and a sufficient condition for the blow-up setto be compact. The proof is postponed to the next section.

24. Blow-up set and space profile 193

Theorem 24.5. Consider problem (22.1) with 1 0

such that, if Ea(u0) < η, then a is not a blow-up point.(ii) Assume in addition that u0(x), |∇u0(x)| → 0 as |x| → ∞, then the blow-up setis compact.

Remarks 24.6. (i) Single-point blow-up. The first example of a single-pointblow-up for problem (22.1) was found in [531] with n = 1 and u0 = kψ, where ψis a positive stationary solution of (22.1) and k, p 1. On the other hand, underthe assumptions of Theorem 24.1 with Ω = R

n instead of a ball, we still have asingle-point blow-up (see [388]; the proof is different from that of Theorem 24.1).

(ii) Blow-up at infinity. By a careful reading of the proof of Theorem 24.5(ii)one obtains the stronger conclusion that sup|u(x, t)| : |x| > R, 0 < t < T < ∞for some large R > 0. On the other hand, the result may fail if no decay is assumedon u0, as shown by the simple example u = k(T − t)−1/(p−1). Furthermore, it hasbeen shown in [249] that, if lim|x|→∞ u0(x) = L > 0 and 0 ≤ u0 ≤ L in R

n, u0 ≡ L,then u remains bounded on compact subsets of R

n up to t = Tmax(u0) < ∞ andblows up only at space infinity (see also [312] for related results). Under the sameassumption, denoting by y the solution of the ODE (23.1) with y(0) = L, itwas also proved in [249] that u and y share the same blow-up time T and thatlim|x|→∞ u(x, t) = y(t), uniformly for t bounded away from T .

(iii) One-dimensional case. Consider problem (22.1) with n = 1 and Ωbounded.

Assume first u0 ≥ 0 and T = Tmax(u0) < ∞. Then the results of [126] guaranteethat B(u0) is finite and its cardinality is bounded above by the number of localmaxima of u0. Moreover, given x /∈ B(u0), there exists ϕ(x) := limt→T u(x, t) andϕ ∈ C2

(Ω \B(u0)

).

On the other hand, given x1, x2, . . . xk ∈ Ω, there exists u0 ≥ 0 such thatB(u0) = x1, . . . , xk, see [364].

The arguments in [209] and the universal bounds in Section 26 show that thereexists T ∗ < ∞ with the following property: If u0 ≥ 0 and Tmax(u0) > T ∗, thenB(u0) consists of a single point.

(iv) Blow-up in the interior. Consider problem (22.1) with Ω bounded andconvex. If u0 ≥ 0 and T = Tmax(u0) < ∞, then B(u0) is always a compactsubset of Ω (see [219]). The idea of the proof is the following: Choose y ∈ ∂Ω.Without loss of generality we may assume that y = 0 and that the hyperplanex = (x1, . . . , xn) : x1 = 0 is tangential to ∂Ω at the origin, with x1 < 0 for allx ∈ Ω. The method of moving planes guarantees ux1 < 0 for all t ≥ η > 0 andx ∈ Σλ := z ∈ Ω : z1 > −λ provided λ > 0 is small enough. Now, similarlyas in the proof of Theorem 24.1 one obtains J := ux1 + ε(x1 − λ)1+δuγ ≤ 0 in

Σλ× (η, T ) which implies u(x) ≤ C(λ− λ)−α for any λ ∈ (0, λ) and x ∈ Σλ (whereα, γ, δ, η, ε have the same meaning as in the proof of Theorem 24.1). This estimateguarantees B(u0) ∩ ∂Ω = ∅.

(v) Global and regional blow-up. Consider problem (14.1) with

f(u) = (1 + u)(log(1 + u)

)b, b > 1.

Assume first that Ω is bounded, u0 ≥ 0 and Tmax(u0) < ∞ (such functions doexist). If b < 2, then the blow-up is global, that is B(u0) = Ω. If b = 2, then theblow-up is either global or regional (that is B(u0) contains a nonempty open set,but B(u0) = Ω), depending on the size of Ω. These results were proved in [313].Similarly, if Ω = R, b = 2 and u0 ≥ 0 is symmetric and radially nonincreasing, u0 ≡0, then Tmax(u0) < ∞, the measure of B(u0) is at least 2π (and B(u0) = [−π, π]under some additional assumptions on u0), see [230]. On the other hand, if Ω isa ball and b > 2, then there are positive initial data such that the correspondingsolutions blow up at a single point (this follows from the proof of Theorem 24.1with the choice F (u) = (1 + u)

(log(1 + u)

)b−1).Regional or global blow-up cannot happen for positive solutions of (18.1) if

p < pS and u0 is continuous, bounded and nonconstant. In this case, the (n− 1)-dimensional Hausdorff measure of B(u0)∩M is finite for any bounded measurableset M ⊂ R

n, see [520]. This is optimal, in view of examples from [246] of solu-tions blowing up on a sphere. Moreover, results on the regularity of B(u0) near anonisolated blow-up point have been obtained in [541], [542].

(vi) Small and large diffusion limits. Consider positive solutions of prob-lem (22.1) with a diffusion coefficient D > 0 in front of the Laplacian, and witheither the Dirichlet or the Neumann boundary conditions. Then, under suitableadditional assumptions, the blow-up set of u concentrates near the maxima of u0

as D → 0. In the limit D → ∞, for the Neumann case, it concentrates near M,whereM is the set of maxima of the L2-projection of u0 onto the second Neumanneigenspace (see [289] and the references therein).

Remark 24.7. Limitations concerning comparison arguments. If Ω is abounded domain, then two ordered sub-/supersolutions cannot share the sameexistence time unless both are global. For instance, if u is the solution of (22.1)with Tmax(u0) < ∞ and if v ≡ u is a supersolution of (22.1) on (0, T ) such thatu0 ≤ v(·, 0) ∈ L∞(Ω), then T < Tmax(u0). This follows from Proposition 27.3below. In particular the knowledge of the blow-up rate or set of v does not providedirect information on that of u (but the situation can be different in unbounded do-mains, cf. the end of Remark 24.6(ii)). Nevertheless, in bounded domains, one cansometimes use indirect comparison arguments (see Proposition 23.1 for a simplecase) or more sophisticated intersection-comparison arguments.

25. Self-similar blow-up behavior 195

25. Self-similar blow-up behavior

In this section we apply the method of similarity variables, introduced in theproof of Theorem 23.7, to study the space-time behavior of solutions of the modelproblem (22.1) near blow-up points as t approaches the blow-up time.

The following theorem is due to [244], [246]. A similar result for n = 1 wasobtained independently in [229]. Theorem 25.1 can be extended to bounded convexdomains (see [246]), but here we restrict ourselves to the case of the whole spacefor simplicity.

Theorem 25.1. Consider problem (22.1) with Ω = Rn, 1 < p ≤ pS, u0 ∈

L∞(Rn), and let k = (p − 1)−1/(p−1). Assume that T := Tmax(u0) < ∞ andthat the upper blow-up rate estimate (23.5) is satisfied. If a is a blow-up point ofu, then we have

limt→T

(T − t)1/(p−1)u(a + y√

T − t, t) = ±k, (25.1)

uniformly on compact sets |y| ≤ C.

Remark 25.2. Let 1 < p < pS , Ω = Rn, u0 ∈ L∞(Rn) and let a be a blow-

up point of u (in the sense of the definition in (24.1)). Theorem 23.7 and Re-mark 23.3(a) guarantee that the upper blow-up rate estimate (23.5) is satisfied.Consequently, it follows in particular from Theorem 25.1 that u does blow-up atx = a and the blow-up rate of u(a, t) is exactly that given by the ODE (cf. (23.1)–(23.2)).

In the proof, among other things, we shall use another result from [246], which isof independent interest since it is valid for any p > 1. If a is a blow-up point, thenthis result provides a lower bound on the blow-up rate. Note that no boundaryconditions are assumed and that this is a purely local result. The proof in [246]was based on a cut-off, the variation-of-constants formula, parabolic estimates andbootstrap. We here give a simpler proof, based on a comparison argument using aquadratic change of unknown and a cut-off (cf. (25.6)).

Theorem 25.3. Let p > 1, T > 0, ρ > 0, a ∈ Rn and denote Q = B(a, ρ) ×

(T − ρ2, T ). There exists ε = ε(n, p) > 0 such that if u is a classical solution of

ut −∆u = |u|p−1u, (x, t) ∈ Q,

and u satisfies|u(x, t)| ≤ ε(T − t)−1/(p−1), (x, t) ∈ Q, (25.2)

then u is uniformly bounded in a neighborhood of (a, T ).

Proof. By a space-time translation, we may assume a = 0 and T = ρ2. By scaling,we may also assume ρ = 1. Indeed, u(x, t) := ρ2/(p−1)u(ρx, ρ2t) solves the same

equation in B1 × (0, 1), and (25.2) is equivalent to |u(x, t)| ≤ ε(1− t)−1/(p−1) for|x| < 1 and t ∈ (0, 1). Set

α = min(1/2, (p− 1)/4). (25.3)

For each R > 0, we may find φ ∈ C2(Rn) such that

φ(x) = 0 for |x| ≥ R/√

2, φ(x) ≥ 1 for |x| ≤ R/2, (25.4)

and|∇φ|2 + |∆φ2| ≤ C(R, n)φ2(1−α) (25.5)

(it suffices to consider φ(x) = (2 − 4R−2|x|2)m+ for m > 2 large enough). Choose

R = 1 and putv = u2φ2. (25.6)

For (x, t) ∈ B1 × (0, T ), we have

vt −∆v = 2uutφ2 − 2φ2(u∆u + |∇u|2)− 8uφ∇u · ∇φ− u2∆φ2.

Since 4|uφ∇u · ∇φ| ≤ φ2|∇u|2 + 4u2|∇φ|2, we deduce that

vt −∆v ≤ 2φ2|u|p+1 + u2(8|∇φ|2 + |∆φ2|). (25.7)

Using (25.5) and assumption (25.2), it follows that

vt −∆v ≤ 2|u|p−1v + u2αv1−αφ−2(1−α)(8|∇φ|2 + |∆φ2|)≤ 2|u|p−1v + Cu2α(1 + v)

≤ 2εp−1(T − t)−1v + Cε2α(T − t)−2α/(p−1)(1 + v).

Here and below, C is a generic positive constant depending only on n, p, whereasK will denote a generic positive constant depending on n, p, u. Assuming 0 < ε < 1and recalling (25.3), we obtain

vt −∆v ≤ Cε2α(T − t)−1v + C(T − t)−2α/(p−1). (25.8)

Let v = (T − t)−Cε2α

+ K(T − t)1−2α/(p−1) for 0 ≤ t < T . We have

vt = Cε2α(T − t)−1v + K(1− 2α/(p− 1)− Cε2α)(T − t)−2α/(p−1).

For ε = ε(n, p) > 0 small and K > 0 large, it follows that v is a supersolution to(25.8) and we also have v(1/2) ≥ ‖v(·, 1/2)‖∞. Since v = 0 on ∂B1 × (0, T ), wededuce from the comparison principle that v ≤ v in B1 × [1/2, T ), hence

u ≤ K(T − t)−Cε2α

, |x| < 1/2, 1/2 ≤ t < T. (25.9)

Now considering v = u2φ2 with R = 1/2 instead of R = 1 in (25.4), and taking asmaller ε(n, p) > 0, inequalities (25.7) and (25.9) imply vt−∆v ≤ K(T − t)−1/2 inB1/2 × (1/2, T ). Using a supersolution of the form K −K(T − t)1/2, we concludethat u is bounded in a neighborhood of (x = 0, t = T ).

Before going into the proof of Theorem 25.1, let us first observe that, consideringthe rescaled solution in similarity variables (cf. (23.7)–(23.8)), the conclusion canbe restated as:

lims→∞ wa(y, s) = ±k, uniformly on compact sets |y| ≤ C.

The basic idea of the proof is to apply dynamical systems arguments to show thatthe global bounded solution wa is attracted by the set of equilibria, i.e. solutionsof

∆z − 12y · ∇z + |z|p−1z − βz = 0, z ∈ R

n (25.10)

(cf. Lemma 25.6(i)). On the other hand, under the assumption p ≤ pS , we shallshow that the only (bounded) equilibria are the constant solutions z = k,−k and0 (Proposition 25.4). The last task will then be to show the nondegeneracy, i.e. toexclude the possibility of wa approaching 0, which will be achieved by combiningTheorem 25.3 with suitable energy arguments (cf. Lemma 25.6(ii)(iii)).

Thus, let us define

S =z ∈ C2 ∩ L∞(Rn) : z is a solution of (25.10)

.

For given a ∈ Rn, we denote

ω(wa) =z ∈ S : ∃sn →∞, wa(y, sn)→ z(y) in C1(Rn).

Proposition 25.4. If 1 pS (provided p < pL defined in (22.18)). In-deed, in that range, Proposition 22.5 shows the existence of backward self-similarsolutions with (positive) bounded nonconstant profile.

(ii) However, if Ω = BR and u ≥ 0 is radial and satisfies ut ≥ 0 and ur ≤ 0,then assertion (25.1) with a = 0 remains true for all p > pS (see [62], and also[358], [359] for further results).

Proof of Proposition 25.4. Let w ∈ S. We first claim that |∇w| is bounded.Indeed, by setting u(x, t) = (1−t)−βw(x/

√1− t), we define a (self-similar) solution

of (18.1) in Rn×(0, 1), with u0 = w ∈ L∞(Rn). Since ∇u(x, 1/2) = 2β+1/2w(

√2x)

and ∇u(·, 1/2) ∈ L∞(Rn) by smoothing effect, the claim follows.


Let us show that w satisfies the Pohozaev-type identity( n

p + 1− n− 2

2

)∫Rn

|∇w|2ρ dy +12

(12− 1

p + 1

)∫Rn

|y|2|∇w|2ρ dy = 0. (25.11)

We shall obtain (25.11) as a linear combination of three other identities. The firstone is ∫

|∇w|2ρ dy + β

∫w2ρ dy −

∫|w|p+1ρ dy = 0. (25.12)

(Here and in what follows all integrals are taken over Rn.) Rewriting (25.10) as

∇ · (ρ∇w) − βρw + ρ|w|p−1w = 0, (25.13)

(25.12) is obtained by multiplying (25.13) by −w and using integration by parts.This procedure can be easily justified since w and |∇w| are bounded and ρ decaysexponentially: It suffices to integrate by parts on BR and then let R → ∞. Thisargument will be used in the rest of the proof without further mention.

The second identity is∫|y|2|∇w|2ρ dy +

∫ [(β +

12

)|y|2 − n

]w2ρ dy −

∫|y|2|w|p+1ρ dy = 0. (25.14)

It is obtained by multiplying (25.13) by −|y|2w and using integration by parts,since

−∫|y|2w∇ · (ρ∇w) dy =

∫|y|2|∇w|2ρ dy +

∫(y · ∇w2)ρ dy

=∫|y|2|∇w|2ρ dy − n

∫w2ρ dy +

12

∫|y|2w2ρ dy.

The third identity is∫ ( |y|24− n− 2

2

)|∇w|2ρ dy +

∫ (β|y|24

− nβ

2

)w2ρ dy

−∫ ( |y|2

2(p + 1)− n

p + 1

)|w|p+1ρ dy = 0.

(25.15)

To get (25.15), we multiply (25.13) by −(y · ∇w) and we use∫(y · ∇w)ρ(βw − |w|p−1w) dy =

∫ρy · ∇

(βw2

2− |w|

p+1

p + 1

)dy

=∫ ( |y|2

2− n)ρ(βw2

2− |w|

p+1

p + 1

)dy


and

−∫

(y · ∇w)∇ · (ρ∇w) dy =∫

(ρ∇w) · ∇(y · ∇w) dy

=∫

ρ|∇w|2 dy +12

∫(ρy) · ∇(|∇w|2) dy

=∫

ρ|∇w|2 dy +12

∫ ( |y|22− n)ρ|∇w|2 dy.

Now, to complete the proof of (25.11), we eliminate the terms involving w2 and|w|p+1 by taking the linear combination n

p+1 ·(25.12) − 12(p+1) ·(25.14) + (25.15).

Finally, (25.11) and our assumption p ≤ pS imply ∇w ≡ 0, hence w ≡ 0, w ≡ kor w ≡ −k.

Lemma 25.6. Consider problem (18.1) with p > 1 and u0 ∈ L∞(Rn). Assumethat the upper blow-up rate estimate (23.5) is satisfied. Then we have:(i) For any sequence sj → ∞, there exists a subsequence, still denoted sj, and afunction z ∈ S such that wa(·, sj) → z in C1(Rn).(ii) Assume that ω(wa) 0 (resp., ±k). Then E

(wa(s)

)→ 0 (resp., E

(wa(s)

)→

η(n, p) > 0).(iii) If ω(wa) 0, then a is not a blow-up point.(iv) If p ≤ pS, then ω(wa) is one of the sets 0, k, −k.

Proof. (i) Assumption (23.5) implies

|wa| ≤ M, y ∈ Rn, s ≥ s0. (25.16)

By Proposition 23.15, since ∇wa(y, s) = (T − t)β+1/2∇u(x, t), this implies

|∇wa| ≤M1, y ∈ Rn, s ≥ s0 := s0 + log 2. (25.17)

Let zj(y, s) = wa(y, s + sj). By (23.9), (25.16), (25.17) and parabolic estimates,the sequence zj is precompact in C2,1(Rn × [0, 1]). Consequently, there exists asubsequence of sj (still denoted sj) and a solution z of

zs −∆z +12z · ∇z = |z|p−1z − βz, y ∈ R

n, s ∈ [0, 1],

such that wa(·, · + sj) → z in C2,1(Rn × [0, 1]). Moreover z,∇z are bounded inR

n × [0, 1]. On the other hand, using (23.17), we have∫ 1

0

∫Rn

(∂szj)2ρ dy ≤∫ ∞

sj

∫Rn

(∂swa)2ρ dy → 0

as j →∞. By Fatou’s lemma we deduce that ∂sz = 0 and the assertion follows.(ii) Assume that wa(·, sj)→ 0 (resp., ±k) and ∇wa(·, sj) → 0, uniformly for |y|

bounded. Using (25.16), (25.17) and dominated convergence, we infer E(wa(sj)

)→

0, resp.

E(wa(sj)

)→(∫

Rn

ρ dy)(β

2− kp−1

p + 1

)k2 =

(4π)n/2k2

2(p + 1)=: η(n, p) > 0.

The assertion then follows from the monotonicity of E(wa(s)

).

(iii) Let b ∈ Rn. Similar to (25.16) and (25.17), we have

|wb| ≤M, |∇wb| ≤ M1, y ∈ Rn, s ≥ s0, (25.18)

with M, M1 independent of b. We shall use the interpolation inequality

|v(0)| ≤ C(n, θ)[‖v‖θ

L2(B1)‖∇v‖1−θ

L∞(B1) + ‖v‖L2(B1)

], v ∈ C1(B1), (25.19)

where 0 < θ < 2/(n + 2) if n ≥ 2 and θ = 1/2 if n = 1. (Inequality (25.19) canbe shown by applying the mean-value theorem to the difference |v(0)|1/(1−θ) −|v(x)|1/(1−θ), integrating over x ∈ B1 and using Holder’s inequality.) Fix θ asabove. By (23.16), since wb exists globally, we have, for any s1 ≥ s0,

‖wb(0, s)‖2L2(B1) ≤ C(n, p)∫

Rn

w2bρ dy ≤ C(n, p)E2/(p+1)(wb(s1)), s ≥ s1.

(25.20)Using (25.18), (25.19) and (25.20), it follows that

|wb(0, s)| ≤ C(n, p)[M1−θ

1 Eθ/(p+1)(wb(s1)) + E1/(p+1)(wb(s1))], s ≥ s1.

Consequently, there exists γ(M1, ε) > 0 such that

E(wb(s1)

)< γ(M1, ε) implies |wb(0, s)| ≤ ε, s ≥ s1.

Assume that ω(wa) 0. Assertion (ii) implies E(wa(s1)

)< γ(M1, ε) for s1 large.

But since, for given s, E(wb(s)

)depends continuously on b (cf. Proposition 23.8),

we infer that E(wb(s1)

)< γ(M1, ε) for |b− a| small. It follows that |wb(y, s)| ≤ ε,

s ≥ s1, hence (T − t)1/(p−1)|u(b, t)| ≤ ε, for (b, t) close to (a, T ). By Theorem 25.3,we conclude that a is not a blow-up point.

(iv) In view of Proposition 25.4, this follows from an obvious connectednessargument.

Proof of Theorem 25.1. This is an immediate consequence of assertions (iii)and (iv) of Lemma 25.6.

As a consequence of the above arguments, we are also now in a position to provethe result stated in the previous section concerning the blow-up set.

Proof of Theorem 24.5. (i) If E(u0) < η, where η is given by Lemma 25.6(ii),then lims→∞ E

(wa(s)

)< η. Consequently, wa cannot converge to ±k, due to

Lemma 25.6(ii). So it converges to 0 and a is not a blow-up point in view ofTheorem 25.1.

(ii) By dominated convergence, under the current assumption on u0, we havelima→∞ E

(wa(s0)

)= 0. The conclusion thus follows from assertion (i).

Remark 25.7. Radial nonincreasing case. In Theorem 25.1, assume in addi-tion that u0 ≥ 0 is radial nonincreasing. Since

u(0, t) = ‖u(t)‖∞ ≥ k(T − t)−1/(p−1)

by Proposition 23.1, the conclusion (with the + sign) follows directly from Lem-ma 25.6(i) and Proposition 25.4, and the nondegeneracy result is not needed.

Remark 25.8. Blow-up profile. The results in Theorems 24.1 and 24.3 con-cerning the blow-up profile in the original variables can be strongly improved, atthe expense of significant technical difficulty, though. In order to do this one hasto linearize problem (23.9) around the nontrivial equilibrium w∞ ≡ k and studythe convergence to w∞ in more detail. Assume Ω = R, u0 ∈ BC(R), u0 ≥ 0 and0 ∈ B(u0). Then one obtains (see [275], [276] and cf. also [207], [208] for earlierresults in that direction) that one of the following alternatives must hold:

1. (T − t)βu(x, t) ≡ k,

2. (T − t)βu(y((T − t)| log(T − t)|

)1/2, t)→ k

(1 + (p− 1)y2/(4p)

)−β as t→ T ,3. there exists C > 0 and an even integer m ≥ 4 such that

(T − t)βu(y(T − t)1/m, t

)→ k(1 + Cym)−β as t→ T,

where the convergence is uniform for y lying in a bounded set. (Note also thatthe second alternative is always true if u0 is symmetric and has a unique localmaximum at x = 0.) As a consequence of this assertion, the following general resultwas obtained in [519]. Assume that u(x, t) is a positive solution of ut − uxx = up

for x ∈ (−R, R) and t ∈ (0, T ) which blows up at t = T . Assume also that itsblow-up set B is contained in [−δ, δ] for some δ < R. Then B is isolated and forany x ∈ B one of the following holds

limx→x

( |x− x|2| log |x− x||

)β

u(x, T ) =( 8p

(p− 1)2)β

,

limx→x

|x− x|mβu(x, T ) = kC−β ,

where β, k, C, m are as above. It is also known that any of the possibilities men-tioned above may happen (see [100], [366]). For results in higher dimensions werefer to [61], [521], [367], [360].

26. Universal bounds and initial blow-up rates

The a priori estimate (22.2) with a universal constant C cannot be true for allglobal solutions of (22.1) for the following reasons. First, such an estimate wouldimply an a priori bound for stationary solutions and we know from Theorem 7.8(ii)that such bound is not true for sign-changing solutions in the subcritical case.Second, we know from Remark 19.12 that there exist nonnegative global classicalsolutions such that ‖u(t)‖∞ →∞ as t → 0+. Anyhow, in the subcritical case, wecan still hope for a universal bound of global nonnegative solutions of (22.1) onthe interval (τ,∞), where τ > 0. In other words, we are interested in the estimate

supt≥τ

‖u(t)‖∞ ≤ C(τ) for all τ > 0. (26.1)

(Note that (26.1) cannot be true in the critical or supercritical case — at least instarshaped domains — due to Theorem 28.7.) It will be natural at the same timeto ask about the dependence of the constant C(τ), as τ → 0.

In fact, this question can be also considered from a different point of view,which gives rise to interesting connections and unifications with questions studiedin Sections 23 and 21. Consider local nonnegative classical solutions of

ut −∆u = up, x ∈ Ω, 0 < t < T,

u = 0, x ∈ ∂Ω, 0 < t < T

(26.2)

(without any prescribed initial conditions). Do there exist estimates of the form

‖u(t)‖∞ ≤ Ct−α, 0 < t ≤ T/2, (26.3)

and‖u(t)‖∞ ≤ C(T − t)−β , T/2 ≤ t < T, (26.4)

where C = C(p, Ω, T ) > 0 is a universal constant, independent of u ? If (26.4) weretrue with β = 1/(p− 1), one would in particular recover the (final) blow-up esti-mates of Section 23, now with a universal constant. Analogously, estimate (26.3)would provide (universal) initial blow-up rates. An interesting question is whatshould be the optimal value of α. Of course, (26.3) or (26.4) implies in particularthe universal bound (26.1) for global nonnegative solutions. Furthermore, we willsee that these estimates are strongly connected with parabolic Liouville-type the-orems and decay of global solutions of the Cauchy problem (see Remark 26.10(i)).

26. Universal bounds and initial blow-up rates 203

The bound (26.1) for all global nonnegative solutions of (22.1) in bounded do-mains was first proved in [200] for p < pBT (note that this exponent has alreadyappeared in an elliptic context in Section 10). As for the initial and final blow-uprate estimates (26.3) and (26.4), they have first been established in [28] for theCauchy problem with p < pF . Those results have been improved and extended ina number of subsequent works, using various techniques. We shall present someof these results and techniques. Some of the proofs rely on rescaling argumentsand apply essentially only to the model problem (26.2), while some others allowto treat nonlinearities f(u) without precise power behavior (see Remarks 26.5 and26.12).

We start with a result whose proof is relatively simple. Better results will begiven later for the model problem (see Theorems 26.6 and 26.8), but the presentapproach, besides its simplicity, has the advantage to be applicable to more generalnonlinearities (see Remark 26.5). It is based on integral bounds obtained by test-function arguments (in particular using the first eigenfunction) and on smoothingproperties in Lq- or Lq

δ-spaces (see Theorem 26.14 below for further results ob-tained by using Lq

δ-spaces).

Theorem 26.1. Assume Ω bounded and 1 0, there existsC(Ω, p, τ) > 0 such that any global nonnegative classical solution of (26.2) satisfies

supt≥τ

‖u(t)‖∞ ≤ C(Ω, p, τ). (26.5)

Remarks 26.2. (i) Instantaneous attractors. In other words, Theorem 26.1(and similar subsequent results) shows the existence of “instantaneous attractors”for global nonnegative trajectories of (26.2). Note that, by standard smoothingeffects, (26.5) guarantees that for each τ > 0, there is a compact (absorbing)subset Kτ of C2(Ω) ∩ C0(Ω), such that any global nonnegative solution of (26.2)remains in Kτ for t ≥ τ (otherwise u has to blow up in finite time).

In terms of the set G+ introduced in Remark 19.12, Theorem 26.1 says that,although G+ itself is unbounded, for each τ > 0, S(τ)G+ is a bounded subset ofL∞(Ω) (where S(t)u0 denotes the solution u(t) of problem (15.1)).

(ii) Differences from equations with absorption. We emphasize that suchlocalization results are of a quite different nature from what occurs in equationswith absorption, such as ut − ∆u + |u|p−1u = 0 with p > 1. Indeed, for thisequation, it is straightforward that all solutions of the Dirichlet or Cauchy problem(with bounded initial data) satisfy the universal estimate ‖u(t)‖∞ ≤ C(p)t−1/(p−1)

for all t > 0. This immediately follows by comparing with the solution y(t) ≡C(p)t−1/(p−1) of the ODE y′ + yp = 0.

In the case of problem (26.2), this is of course not true, due to the existence ofblowing-up solutions. The universal bound (26.5) is verified by a solution u, under

the assumption that u exists globally (or on some time interval (0, T ) in the caseof estimates (26.3) and (26.4)).

We give a first proof of Theorem 26.1 based on Lqδ-spaces, due to [200]. We first

derive some basic estimates for positive solutions of (26.2).

Lemma 26.3. Assume Ω bounded, p > 1, and 0 < T < ∞. Let u be a nonnegativeclassical solution of (26.2) on (0, T ). Then for all t ∈ (0, T/2], there holds∫

Ω

u(t)ϕ1 dx ≤ C(p, Ω)(1 + T−1/(p−1)), (26.6)

and ∫ t

0

∫Ω

upϕ1 dx ds ≤ C(p, Ω)(1 + t

)(1 + T−1/(p−1)

). (26.7)

Proof. As in the proof of Theorem 17.1, denote y = y(t) :=∫Ω

u(t)ϕ1 dx, multiplythe equation in (26.2) by ϕ1 and integrate by parts. We obtain

d

dt

∫Ω

u(t)ϕ1 dx + λ1

∫Ω

u(t)ϕ1 dx =∫

Ω

up(t)ϕ1 dx. (26.8)

By Jensen’s inequality, we infer that

d

dt

∫Ω

u(t)ϕ1 dx ≥(∫

Ω

u(t)ϕ1 dx

)p

− λ1

∫Ω

u(t)ϕ1 dx.

Since u exists on (0, T ), we deduce easily that∫Ω

u(t)ϕ1 dx ≤ C(p, Ω)(1 + (T − t)−1/(p−1)), 0 < t < T,

hence (26.6). Integrating (26.8) in time over (τ, t) (0 < τ < t ≤ T/2) and using(26.6), we obtain∫ t

τ

∫Ω

upϕ1 dx ds = λ1

∫ t

τ

∫Ω

uϕ1 dx ds +∫

Ω

u(t)ϕ1 dx−∫

Ω

u(τ)ϕ1 dx

≤ C(p, Ω)(1 + t

)(1 + T−1/(p−1)

)and (26.7) follows by letting τ → 0.

Proof of Theorem 26.1. By Theorem 22.1, we know that global solutions of(22.1) satisfy the a priori estimate

‖u(t)‖∞ ≤ C(Ω, p, ‖u(t0)‖∞), t ≥ t0 ≥ 0,

where C remains bounded for ‖u(t0)‖∞ bounded. Therefore, it is sufficient to showthe existence of C(p, Ω, τ) > 0 such that any global classical solution u of (26.2)satisfies

inft∈(0,τ)

‖u(t)‖∞ ≤ C(p, Ω, τ). (26.9)

Moreover, by the Lqδ-smoothing estimate in Theorem 15.9, (26.9) will follow if we

can show that, for some q > (n + 1)(p− 1)/2,

inft∈(0,τ/2)

‖u(t)‖q,δ ≤ C(p, Ω, q, τ). (26.10)

But (26.7) guarantees that (26.10) is true for q = p and, since p < pBT , we havep > (n + 1)(p− 1)/2.

We now give a second proof (see [200, Section 6]), which does not use Lqδ-spaces.

Instead it requires the following estimate, whose proof uses the special test-functionconstructed in Lemma 10.4 by considering a singular elliptic problem.

Lemma 26.4. Assume Ω bounded, p > 1, 0 < T < ∞, and ε ∈ (0, (p + 1)/2]. Letu be a nonnegative classical solution of (26.2) on (0, T ). Then for all t ∈ (0, T/2],there holds ∫ t

0

∫Ω

up+12 −ε dx ds ≤ C(p, Ω, ε)

(1 + t

)(1 + T−1/(p−1)

).

Proof. For given 0 < α < 1, Lemma 10.4 ensures the existence of a functionξ ∈ C(Ω) ∩ C2(Ω) ∩H1

0 (Ω) such that −∆ξ = ϕ−α1 in Ω. Moreover, ξ satisfies

ξ(x) ≤ C(Ω, α)δ(x), x ∈ Ω. (26.11)

Here we choose α = 1 − 4εp−1+2ε . Taking ξ as a test-function in (26.2) (which is

possible due to ξ ∈ H10 (Ω)) and integrating in time over (τ, t), we obtain∫ t

τ

∫Ω

uϕ−α1 dx ds =

∫ t

τ

∫Ω

upξ dx ds +∫

Ω

u(τ)ξ dx−∫

Ω

u(t)ξ dx.

Due to (26.11), (26.6) and (26.7) readily imply∫ t

0

∫Ω

uϕ−α1 dx ds ≤ C(p, Ω, ε)

(1 + t

)(1 + T−1/(p−1)

).

Using Holder’s inequality, the last estimate and (26.7) imply the lemma.

Second proof of Theorem 26.1. As in the first proof, it is sufficient to showthe existence of C(p, Ω, τ) > 0 such that any global classical solution u of (26.2)satisfies (26.9). Moreover, by the smoothing estimate in Theorem 15.2, (26.9) willfollow if we can show that, for some q > n(p− 1)/2,

inft∈(0,τ/2)

‖u(t)‖q ≤ C(p, Ω, q, τ). (26.12)

But Lemma 26.4 guarantees that (26.12) is true for all q ∈ [1, (p+1)/2) and, sincep < pBT , we have q > n(p− 1)/2 for q < (p + 1)/2 close to (p + 1)/2.

Remark 26.5. The assumption p < pBT in Theorem 26.1 is not optimal for themodel problem (26.2), see Theorems 26.6 and 26.8 below. However, unlike theproofs of those theorems, the proof of Theorem 26.1 does not rely on rescalingand can be applied to more general nonlinearities f(x, u) satisfying C1u

q − C ≤f(x, u) ≤ C2u

p + C with p < pBT , under suitable assumption on q ∈ (1, p) (see[450]). Note that the proof uses a priori estimates of global solutions obtained inTheorem 22.1. However, the proof of Theorem 22.1 based on interpolation can bealso extended to such nonlinearities.

Now we give an optimal result [438] in dimensions n ≤ 3 concerning univer-sal bounds of global nonnegative solutions of the Dirichlet problem. The methodis completely different. It is based on energy, measure arguments, rescaling andelliptic Liouville-type theorems.

Theorem 26.6. Let n ≤ 3 and 1 0 such that any global classical solution u of (26.2) satisfies (26.9).Moreover, since p + 1 > n(p− 1)/2 due to p < pS , by the smoothing property inTheorem 15.2, (26.9) will follow if we can show that

inft∈(0,τ/2)

‖u(t)‖p+1 ≤ C(p, Ω, τ).

We argue by contradiction and assume that for each k = 1, 2, . . . , there existsa global solution uk ≥ 0 of (26.2) such that

‖uk(t)‖p+1p+1 > k for all t ∈ (0, τ/2). (26.13)

Denote

Ek(t) = E(uk(t)

)=

12

∫Ω

|∇uk(t)|2 dx− 1p + 1

∫Ω

up+1k (t) dx.

Recall that E′k(t) = −‖∂tuk(t)‖22 ≤ 0 and that uk satisfies the identity

12

d

dt

∫Ω

u2k(t) dx =

∫Ω

up+1k (t) dx−

∫Ω

|∇uk(t)|2 dx

= −2Ek(t) +p− 1p + 1

∫Ω

up+1k (t) dx.

(26.14)

We now proceed in several steps. From now on, C will denote a positive constantand k0 a positive integer, both depending only on p, Ω, τ (and also on q in Steps 4and 5).

Step 1. We claim thatEk(τ/4) ≥ k1/2, (26.15)

for all k ≥ k0 large enough.Assume (26.15) is false. Using (26.14), E′

k ≤ 0 and Holder’s inequality, weobtain, for all t ≥ τ/4,

12

d

dt

∫Ω

u2k(t) dx ≥ −2k1/2 +

p− 1p + 1

∫Ω

up+1k (t) dx, (26.16)

hence12

d

dt

∫Ω

u2k(t) dx ≥ −2k1/2 + C

(∫Ω

u2k(t) dx

)(p+1)/2

.

This implies ∫Ω

u2k(t) dx ≤ Ck

1p+1 , t ≥ τ/4, (26.17)

since otherwise∫Ω u2

k(t) dx has to blow up in finite time. Integrating (26.16) over(τ/4, τ/2) and using (26.13), (26.17), we obtain

14kτ ≤

∫ τ/2

τ/4

∫Ω

up+1k dx dt ≤ C(k

1p+1 + k1/2τ),

a contradiction for k ≥ k0 large.Step 2. Let a > 0 to be fixed later and set Fk = t ∈ (0, τ/4] : −E′

k(t) ≥E

1+1/ak (t). We claim that |Fk| < τ/8 for all k ≥ k0 large enough.Note that Ek > 0 on (0, τ/4] for k ≥ k0 by (26.15), since E′

k ≤ 0. By definitionof Fk, it follows that

(aE−1/ak )′ = −E′

kE−1−1/ak ≥ χFk

on (0, τ/4].

By integration, we deduce that aE−1/ak (τ/4) ≥ |Fk|. The claim then follows from

(26.15).Step 3. Choose

a ≥ (p + 1)/(p− 1). (26.18)

We claim that for all k ≥ k0 large,

‖∂tuk(t)‖22 ≤ C(∫

Ω

up+1k (t) dx

)(a+1)/a

for all t ∈ (0, τ/4] \ Fk. (26.19)

For all t ∈ (0, τ/4] \ Fk, we have

‖∂tuk(t)‖22 = −E′k(t) ≤ E

1+1/ak (t) ≤ ‖∇uk(t)‖2(1+1/a)

2 . (26.20)

Hence, by (26.14) as well as Holder’s and Young’s inequalities,

‖∇uk(t)‖22 ≤∫

Ω

up+1k (t) dx + ‖uk(t)‖2‖∂tuk(t)‖2

≤∫

Ω

up+1k (t) dx + ‖uk(t)‖2‖∇uk(t)‖1+1/a

2

≤∫

Ω

up+1k (t) dx + C‖uk(t)‖p+1‖∇uk(t)‖1+1/a

2

≤∫

Ω

up+1k (t) dx + C‖uk(t)‖2a/(a−1)

p+1 +12‖∇uk(t)‖22

≤ C

∫Ω

up+1k (t) dx +

12‖∇uk(t)‖22,

where we have used (26.18) and (26.13). Consequently,

‖∇uk(t)‖22 ≤ C

∫Ω

up+1k (t) dx.

This along with (26.20) implies (26.19).Step 4. Let 0 < q < (p + 1)/2, b = (p + 1− q)(a + 1)/a and

Gk =t ∈ (0, τ/4] : ‖∂tuk(t)‖22 ≤ C‖uk(t)‖b

∞.

We claim that |Gk| > 0.Due to Lemma 26.4, for A = A(p, q, Ω, τ) > 0 large enough, the set

Gk := t ∈ (0, τ/4] :∫

Ω

uqk(t) dx ≥ A

satisfies|Gk| < τ/8. (26.21)

We deduce from (26.13) that, for all t ∈ (0, τ/4] \ Gk,∫Ω

up+1k (t) dx ≤ C‖uk(t)‖p+1−q

∞

∫Ω

uqk(t) dx ≤ C‖uk(t)‖p+1−q

∞ .

Therefore, Gk ⊃ (0, τ/4]\ (Fk ∪ Gk) by Step 3. The claim then follows from Step 2and (26.21).

Step 5. We will now obtain a contradiction by using a rescaling argument.By Step 4, for each large k, we may pick tk ∈ Gk. By (26.13), we have Mk :=

‖uk(tk)‖∞ → ∞. Choose xk ∈ Ω such that uk(xk, tk) = Mk, denote νk =M

−(p−1)/2k and put

wk(y) = M−1k uk(xk + νky, tk),

wk(y) = M−pk ∂tuk(xk + νky, tk).

Then the functions wk, wk satisfy

∆wk + wpk = wk in Ωk,

wk = 0 on ∂Ωk,

(26.22)

where Ωk = ν−1k (Ω − xk). Moreover, 0 ≤ wk ≤ 1 = wk(0). Now passing to the

limit we will obtain a contradiction in the same way as in [241]; we only have toshow that the functions wk are (locally) uniformly Holder continuous and wk → 0in an appropriate way.

Hence let R > 0, BR(x0) = x ∈ Ω : |x−x0| < R and BkR = y ∈ Ωk : |y| < R.

Since tk ∈ Gk, we have∫Bk

R

|wk(y)|2 dy = M−2pk

∫Bk

R

|∂tuk(xk + νky, tk)|2 dy

= M−2pk ν−n

k

∫BRνk(xk)

|∂tuk(x, tk)|2 dx

≤ CM−2pk M

n(p−1)/2k M b

k = CMγk

for k ≥ k0, where

γ = −2p +a + 1

a(p + 1− q) +

n(p− 1)2

.

By taking q close to (p+1)/2 and a sufficiently large, γ will be negative provided(n− 3)p < n− 1. (In particular this is true due to p < pS if n ≤ 4.)

Consequently, ∫Bk

R

|wk(y)|2 dy → 0

for any R > 0. Since 0 ≤ wk ≤ 1 and wk solves (26.22), standard regularity theoryguarantees that wk is uniformly bounded in W 2,2(Bk

R). Since W 2,2 is embedded inthe space of Holder continuous functions due to n ≤ 3, we may pass to the limitin (26.22), similarly as in the proof of Theorem 12.1, in order to get a limitingsolution w ≥ 0 satisfying the equation ∆w+wp = 0 either in R

n or in a half-space(and satisfying the homogeneous Dirichlet boundary conditions in the latter case).Moreover w ≤ 1 and w(0) = 1, which contradicts the Liouville-type Theorems 8.1and 8.2.

Remark 26.7. By a (nontrivial) modification of the proof of Theorem 26.6, onecan show that the result remains true for n = 4, and for n ≥ 5 under the strongerrestriction p < (n− 1)/(n− 3) < pS (see [450]).

We now turn to universal initial and final blow-up rates. Recall that the expo-nent pB in (26.23) has appeared in Section 21.

Theorem 26.8. Let p > 1, T > 0 and u be a nonnegative classical solution of(26.2) on QT . Assume that either

p < pB, or p < pS, Ω = Rn or Ω = BR, and u radial. (26.23)

Then there holds

u(x, t) ≤ C(n, p)(t−1/(p−1) + (T − t)−1/(p−1)

), x ∈ R

n, 0 < t < T (26.24)

if Ω = Rn, and

u(x, t) ≤ C(p, Ω)(1 + t−1/(p−1) + (T − t)−1/(p−1)

), x ∈ Ω, 0 < t < T (26.25)

otherwise.

Estimates (26.24), (26.25) provide a universal localization in L∞(Ω) throughoutthe time interval (0, T ) for positive trajectories of (26.2). Note that Theorem 26.8partially improves the above results on universal bounds of global solutions to theDirichlet problem. Theorem 26.8 in the case Ω = R

n with p < pB is due to [79],where it follows from integral estimates for local solutions (cf. Proposition 21.5).The other cases are due to [425] and the proof is based on a doubling lemma,a rescaling argument, and the parabolic Liouville-type theorems established inSection 21. The methods are thus different from those in Theorems 26.1 and 26.6.

As an interesting consequence of Theorem 26.8, one obtains the decay of allnonnegative global solutions to the Cauchy problem.

Theorem 26.9. Let p > 1 and u be a global nonnegative classical solution of

ut −∆u = up, x ∈ Rn, t > 0. (26.26)

Assumep < pB, or p < pS and u radial.

Then there holds

u(x, t) ≤ C(n, p) t−1

p−1 , x ∈ Rn, t > 0. (26.27)

Remarks 26.10. (i) If the parabolic Liouville-type Theorem 21.2 were knownfor all p < pS , then this would imply Theorems 26.8 and 26.9 for all p < pS aswell. Conversely, it is clear that estimate (26.24) or (26.27) implies nonexistence ofpositive solutions of (21.1). We see that Liouville-type theorems and these universalestimates are thus equivalent. On the other hand, Theorem 26.8 guarantees thatTheorem 21.1 remains true for nontrivial nonnegative radial classical solutions,bounded or not, and that Theorem 21.2 remains true for nontrivial nonnegativeclassical solutions.

(ii) For all p < pS and without radial symmetry assumption, it is however knownthat the solution of the Cauchy problem (18.1) satisfies ‖u(t)‖∞ → 0 as t → ∞(without a universal estimate), provided u is global and 0 ≤ u0 ∈ L∞ ∩ L2(Rn);see [478] and cf. also [299].

(iii) In Theorems 26.8 and 26.9, no conditions at space infinity are assumed onthe solution u.

(iv) Consider problem (26.2) with the nonlinearity up replaced by f(u). Assumethat f : [0,∞)→ R is continuous and is such that lims→∞ s−pf(s) exists in (0,∞).Then Theorem 26.8 remains valid (with C in (26.24)-(26.25) depending also onf and with an additive constant 1 in estimate (26.24) as well). If we assume inaddition that f is C1 and verifies |f ′(s)| ≤ C(1 + sp−1), s ≥ 0, then Theorem 26.6remains valid.

(v) When Ω is a convex bounded domain, estimate (26.4) with β = 1/(p−1) andC = C(p, Ω, T ) is known also for p < pS , n ≤ 4 [450]. This follows by combiningTheorem 26.6 (cf. also Remark 26.7) with the a priori estimate of the blow-up rate(cf. Remark 23.9). Let us point out that the method of proof of Theorem 26.6 canbe modified to establish initial blow-up rate estimates as well [450], but the valuesof α = α(n, p) obtained in this way are not optimal.

We will use the following key doubling lemma [424].

Lemma 26.11. Let (X, d) be a complete metric space and let ∅ = D ⊂ Σ ⊂ X,with Σ closed. Set Γ = Σ \D. Finally let M : D → (0,∞) be bounded on compactsubsets of D and fix a real k > 0. If there exists y ∈ D such that

M(y) dist(y, Γ) > 2k, (26.28)

then there exists x ∈ D such that

M(x) dist(x, Γ) > 2k, M(x) ≥ M(y), (26.29)

andM(z) ≤ 2M(x) for all z ∈ D ∩ BX

(x, k M−1(x)

).

Proof. Assume that the lemma is not true. Then we claim that there exists asequence (xj) in D such that

M(xj) dist(xj , Γ) > 2k, (26.30)

M(xj+1) > 2M(xj), (26.31)

andd(xj , xj+1) ≤ kM−1(xj) (26.32)

for all j ∈ N. We choose x0 = y. By our contradiction assumption, there existsx1 ∈ D such that

M(x1) > 2M(x0)

andd(x0, x1) ≤ k M−1(x0).

Fix some i ≥ 1 and assume that we have already constructed x0, . . . , xi such that(26.30)–(26.32) hold for j = 0, . . . , i− 1. We have

dist(xi, Γ) ≥ dist(xi−1, Γ)− d(xi−1, xi) > (2k − k)M−1(xi−1) > 2k M−1(xi),

henceM(xi) dist(xi, Γ) > 2k.

By our contradiction assumption, it follows that there exists xi+1 ∈ D such that

M(xi+1) > 2M(xi)

andd(xi, xi+1) ≤ k M−1(xi).

We have thus proved the claim by induction.Now, we have

M(xi) ≥ 2iM(x0) and d(xi, xi+1) ≤ k 2−iM−1(x0), i ∈ N. (26.33)

In particular, (xi) is a Cauchy sequence, hence it converges to some a ∈ D ⊂ Σ.Moreover,

d(x0, xi) ≤i−1∑j=0

d(xj , xj+1) ≤ k M−1(x0)i−1∑j=0

2−j ≤ 2k M−1(x0),

hencedist(xi, Γ) ≥ dist(x0, Γ)− 2k M−1(x0) =: δ > 0.

Therefore, K := xi : i ∈ N ∪ a is a compact subset of Σ \ Γ = D. SinceM(xi) → ∞ as i → ∞ by (26.33), this contradicts the assumption that M isbounded on compact subsets of D. The lemma is proved.

Proof of Theorem 26.8. We first consider the nonradial case and assume p <pB. We will show (26.25). Note that if Ω = R

n, by a simple scaling argument(replacing u(x, t) by u(y, s) := T 1/(p−1)u(

√Ty, T s)), (26.25) with T = 1, implies

(26.24) for any T > 0.

Assume that estimate (26.25) fails. Then, there exist sequences Tk ∈ (0,∞), uk,yk ∈ Ω, sk ∈ (0, Tk), such that uk solves (26.2) (with T replaced by Tk) and thefunctions

Mk := up−12

k , k = 1, 2, . . . , (26.34)

satisfyMk(yk, sk) > 2k (1 + d−1

k (sk)), (26.35)

where dk(t) := (min(t, Tk − t))1/2. We will use Lemma 26.11 with X = Rn+1,

equipped with the parabolic distance

dP

((x, t), (y, s)

)= |x− y|+ |t− s|1/2,

Σ = Σk = Ω × [0, Tk], D = Dk = Ω × (0, Tk), and Γ = Γk = Ω × 0, Tk. Noticethat

dk(t) = distP

((x, t), Γk

), (x, t) ∈ Σk.

By Lemma 26.11, it follows that there exists xk ∈ Ω, tk ∈ (0, Tk) such that

Mk(xk, tk) > 2k d−1k (tk), (26.36)

Mk(xk, tk) ≥Mk(yk, sk) > 2k,

andMk(x, t) ≤ 2Mk(xk, tk), (x, t) ∈ Dk ∩ Bk, (26.37)

whereBk :=

(x, t) ∈ R

n+1 : |x− xk|+ |t− tk|1/2 ≤ k λk

,

andλk := M−1

k (xk, tk)→ 0. (26.38)

Observe that for all (x, t) ∈ Bk, we have |t−tk| ≤ k2λ2k < d2

k(tk) = min(tk, Tk−tk)by (26.36), hence t ∈ (0, Tk). It follows that(

Ω ∩ |x− xk| < kλk

2 )× (tk − k2λ2

k

4 , tk + k2λ2k

4 ) ⊂ Dk ∩ Bk.

Now we rescale uk by setting

vk(y, s) := λ2/(p−1)k uk(xk + λky, tk + λ2

ks), (y, s) ∈ Dk, (26.39)

whereDk :=

(λ−1

k (Ω− xk) ∩ |y| < k/2)× (−k2/4, k2/4).

The function vk solves

∂svk −∆yvk = vpk, (y, s) ∈ Dk,

vk = 0, y ∈ λ−1k (∂Ω− xk), |y| < k/2, |s| < k2/4.

(26.40)

Moreover we have vk(0, 0) = 1 and (26.37) implies

vk ≤ C := 22

p−1 , (y, s) ∈ Dk. (26.41)

Let ρk := dist(xk, ∂Ω). By passing to a subsequence, we may assume that either

ρk/λk →∞, (26.42)

orρk/λk → c ≥ 0. (26.43)

In case (26.42) holds, by using (26.40), (26.41), (26.38), interior parabolic esti-mates and the embedding (1.2), we deduce that some subsequence of vk convergesin Cα(Rn+1), 0 < α < 1, to a bounded classical solution u ≥ 0 of (21.1) withu(0, 0) = 1. Moreover, as a consequence of the strong maximum principle, we haveeither u > 0 in R

n+1, or

u = 0 in Rn × (−∞, t0] and u > 0 in Q := R

n × (t0,∞), (26.44)

for some t0 < 0. But, in the latter case, since u ≤ C, we have ut−∆u ≤ Cp−1u inQ and we infer from the maximum principle in Proposition 52.4 that u = 0 in Q,a contradiction. Therefore u > 0, which contradicts Theorem 21.1.

In case (26.43) holds, denote Hc := y ∈ Rn : y1 > −c. By performing a suit-

able orthogonal change of coordinates, similarly as in the proof of Theorem 12.1, us-ing (26.38), (26.40), (26.41), interior-boundary parabolic estimates and the embed-ding (1.2), we obtain a subsequence of vk which converges in Cα(Hc), 0 < α < 1,to a bounded classical solution v ≥ 0 of

∂sv −∆yv = vp, y ∈ Hc, s ∈ R,

v = 0, y ∈ ∂Hc, s ∈ R,

(26.45)

with v(0, 0) = 1 (hence c > 0). Similarly as in the previous case, we obtain v > 0,which contradicts Theorem 21.8.

In the radial case, let us assume in addition that u(|x|, t) is nonincreasing as afunction of |x|. Then we may take xk = 0 in the above proof and the rescaling pro-cedure yields a positive, bounded, radial, classical solution of (21.1), contradictingthe radial Liouville-type Theorem 21.2. For the general (nonmonotone) radial case,which is slightly more delicate, we refer to [425].

Remark 26.12. Lemma 26.11 and the method of proof of Theorem 26.8 are ageneralization of an idea in [283] (see also, e.g., [130], [448], [361], [339]). In thoseworks, blow-up estimates and a priori bounds of global solutions, with nonuni-versal constants, were derived for various types of superlinear parabolic problems.By using a property similar to Lemma 26.11 (but concerning functions of the

time variable only), it was shown that if a solution u were violating a suitableestimate, then the function M(t) := ‖u(t)‖∞ would satisfy M(s) ≤ 2M(tk) forall s ∈ [tk, tk + kM1−p(tk)] and some sequence of times tk. Then, by a rescalingargument similar to that used in the proof of Theorem 26.8, one was led to acontradiction with a Fujita-type theorem. Note that these approaches do not useany variational structure of the problem, unlike the methods in the proofs of The-orems 22.1 and 23.7 for instance. This advantage will be exploited in Sections 38and 44.

A natural question is whether the exponent 1/(p−1) in Theorem 26.8 is optimal.As for the (final) blow-up rates, this is indeed the case, due to Proposition 23.1.Interestingly, the situation is different for the initial blow-up rate, as it appearsfrom the following results, which show that for p close to 1, the optimal initialblow-up rate exponents are in fact less than 1/(p−1). Moreover, they are differentfor the Cauchy and for the Dirichlet problems.

Theorem 26.13. Let p > 1, T > 0, and Ω = Rn.

(i) Assume p < pF . Then any nonnegative classical solution of (26.2) on Rn×(0, T )

satisfiesu(x, t) ≤ C(n, p, T ) t−n/2, x ∈ R

n, 0 < t < T/2.

(ii) Let

α0 := min(n

2,

1p− 1

).

For all ε > 0, there exist T > 0, a positive classical solution u of (26.2), and C > 0such that

‖u(t)‖∞ ≥ Ct−α0+ε, for t > 0 small.

Theorem 26.14. Let p > 1, T > 0, and assume Ω bounded.

(i) Assume p < 1 + 2/(n + 1). Then any nonnegative classical solution of (26.2)on QT satisfies

u(x, t) ≤ C(p, Ω, T )t−(n+1)/2, x ∈ Ω, 0 < t < T/2.

(ii) Let

α1 := min(n + 1

2,

1p− 1

).

For all ε > 0, there exist T > 0, a positive classical solution u of (26.2), and C > 0such that

‖u(t)‖∞ ≥ Ct−α1+ε, for t > 0 small.

Remarks 26.15. (i) The proof of Theorem 26.13 yields

C(n, p, T ) = C(n, p)T n/2−1/(p−1).

(ii) As already mentioned in Remark 15.4(ii), it is known [269] that if pF < p <pS , then (26.26) possesses global, positive self-similar solutions of the form u(x, t) =t−1/(p−1)w(|x|/

√t), with w ∈ C2([0,∞)), radial and decreasing. In particular we

have ‖u(t)‖∞ = w(0)t−1/(p−1), t > 0 (compare with Theorem 26.9).(iii) By minor modifications of the proof, one can show that Theorem 26.14(i)

remains valid for more general nonlinearities f(u) instead of up, see [450]. Namelyone may assume that f , of class C1, satisfies C1s

q−C2 ≤ f(s) ≤ C2(1+sp), s ≥ 0,for some 1 < q 0. A similar generalization istrue for Theorem 26.13(i).

Theorem 26.13(i) was proved in [79] by using Harnack inequality for the linearparabolic equation ut − ∆u = V (x, t)u, which holds under suitable integrabilityconditions on the potential V . An alternative proof relying on local regularity esti-mates from [29] (based on Moser’s iteration arguments) was also given in [79]. Herewe provide a more elementary proof (based on a modification of ideas from [361]),which relies on smoothing in uniformly local Lebesgue spaces (cf. Section 15). Theintroduction of these spaces in our problem is natural. Indeed, a simple applicationof the eigenfunction method (cf. Section 17) provides the following uniformly localL1 a priori estimate.

Lemma 26.16. Let u be a nonnegative classical solution of ut − ∆u = up inR

n × (0, T ). Then there holds

‖u(t)‖1,ul = supa∈Rn

∫|y−a|<1

|u(y, t)| dy ≤ C(n, p)(1 + T−1/(p−1)), 0 < t < T/2.

(26.46)

Proof. Let ϕ1 be the first positive eigenfunction of −∆ in the ball B2 ⊂ Rn, with

zero Dirichlet conditions. As usual, we normalize ϕ1 by∫

B2ϕ1 = 1 and denote by

λ1 the corresponding eigenvalue. Multiplying the equation by ϕ1, integrating byparts over B2, using ∂νϕ1 ≤ 0 on ∂B2 and Jensen’s inequality, we obtain

d

dt

∫B2

u(t)ϕ1 dx ≥∫

B2

up(t)ϕ1 dx− λ1

∫B2

u(t)ϕ1 dx

≥(∫

B2

u(t)ϕ1 dx)p

− λ1

∫B2

u(t)ϕ1 dx

for all 0 < t < T . By a standard differential inequality argument, it follows that∫B1

u(t) dx ≤ C(n)∫

B2

u(t)ϕ1 dx ≤ C(n, p)(1 + T−1/(p−1)), 0 < t < T/2.

The estimate then follows by applying this to u(x−a, t) and taking the supremumover a ∈ R

n.

Proof of Theorem 26.13. (i) By a simple scaling argument (replacing u(x, t)by u(y, s) := T 1/(p−1)u(

√Ty, T s)), it is enough to show the estimate for T = 1.

By Lemma 26.16, we have

‖u(t)‖1,ul ≤ C(n, p), 0 < t < 1/2.

We may then apply uniformly local Lq-smoothing results as follows. Fix t ∈ (0, 1/2]and let t1, t2 > 0 be such that t = t1 + t2. Since p < pF , we have 1 > n(p− 1)/2.Moreover, due to Theorem 26.8, we have u ∈ L∞

loc((0, 1), L∞(Rn)). It then followsfrom Theorem 15.11 with q = 1 and (26.46) that

‖u(t)‖∞ ≤ C(n, p)‖u(t1)‖1,ul t−n/22 ≤ C(n, p) t

−n/22 ,

provided t2 ≤ τ = τ(n, p).If t ≤ τ , we take t1 = t2 = t/2. If τ < t ≤ 1/2, we take t2 = τ , t1 = t − τ . In

both cases, we thus obtain

‖u(t)‖∞ ≤ C(n, p)t−n/2.

(ii) Let q ∈ (q0,∞) with q0 := max(1, n(p− 1)/2). By Theorem 15.2, we knowthat problem (22.1) is locally well-posed in Lq(Rn). Let u0(x) = |x|−2kχ|x|<1,with k < n/2q0. Then u0 ∈ Lq(Rn) for q > q0 close to q0 and, by estimate (15.30),we have ‖e−tAu0‖∞ ≥

(e−tAu0

)(0) = Ct−k for t > 0 small. It follows that the

local solution u of (22.1) with initial data u0 satisfies ‖u(t)‖∞ ≥ Ct−k for smallt > 0. Since k → α0 as k → n/2q0, the conclusion follows.

The proof of Theorem 26.14(i) is similar to that of Theorem 26.13(i), exceptthat we now use smoothing in Lq

δ-spaces (cf. Section 15).

Proof of Theorem 26.14(i). Due to (1.4), estimate (26.6) can be restated asan L1

δ-estimate:

‖u(t)‖1,δ ≤ M := C(p, Ω)(1 + T−1/(p−1)), 0 < t ≤ T/2. (26.47)

We can now apply Lqδ-smoothing results as follows. Fix t ∈ (0, T/2] and let t1, t2 >

0 be such that t = t1 + t2. Since 1 > (n + 1)(p − 1)/2, due to p < 1 + 2/(n + 1),we may apply Theorem 15.9 with q = 1, and we deduce from (26.47) that

‖u(t)‖∞ ≤ C(p, Ω, M) t−(n+1)/22 ,

provided t2 ≤ τM := τM (p, Ω, M).

If t ≤ τM , we take t1 = t2 = t/2. If τM < t ≤ T/2, we take t2 = τM , t1 = t−τM ,and we note that t

−(n+1)/22 ≤ ( T

τM)(n+1)/2t−(n+1)/2. In both cases we thus obtain

‖u(t)‖∞ ≤ C(p, Ω, T )t−(n+1)/2, 0 < t ≤ T/2.

(ii) Let q ∈ (q1,∞) with q1 := max(1, (n + 1)(p − 1)/2). By Theorem 15.9, weknow that problem (22.1) is locally well-posed in Lq

δ(Ω). By Theorem 49.7(ii) inAppendix C, for any k < (n + 1)/2q, there exists u0 ∈ Lq

δ(Ω) such that

‖e−tAu0‖∞ ≥ Ct−k

for small t > 0. It follows that the local solution u of (22.1) with initial data u0

satisfies ‖u(t)‖∞ ≥ Ct−k for small t > 0. Since k → α1 as k → (n + 1)/2q1, theconclusion follows.

27. Complete blow-up

In this section we consider the question whether or not nonglobal classical solutionsof problem (22.1) can be continued in some weak sense after the blow-up timeTmax(u0).

A natural way to look at this question is via monotone approximation. Let ube the solution of problem (22.1) and assume u0 ≥ 0 and Tmax(u0) <∞. Set

fk(v) := min(vp, k), v ≥ 0, k = 1, 2, . . .

and let uk be the solution of the problem

vt −∆v = fk(v), x ∈ Ω, t > 0,

v = 0, x ∈ ∂Ω, t > 0,

v(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (27.1)

The function uk is globally defined and uk+1 ≥ uk. Define

u(x, t) := limk→∞

uk(x, t).

Notice that uk solves the integral equation

uk(x, t) =∫

Ω

G(x, y, t)u0(y) dy +∫ t

0

∫Ω

G(x, y, t− s)fk(uk(y, s)) dy ds,

x ∈ Ω, t > 0,

(27.2)

27. Complete blow-up 219

where G is the Dirichlet heat kernel in Ω. Since G > 0 and uk+1 ≥ uk, we maypass to the limit in (27.2) in order to get

u(x, t) =∫

Ω

G(x, y, t)u0(y) dy+∫ t

0

∫Ω

G(x, y, t−s)up(y, s) dy ds, x ∈ Ω, t > 0,

(27.3)where the double integral may be infinite. Obviously u(·, t) = u(·, t) for t <Tmax(u0). Set

T c = T c(u0) := inft ≥ Tmax(u0) : u(x, t) = ∞ for all x ∈ Ω

and notice that T c(u0) ≥ Tmax(u0). Moreover, due to (27.3) and

uk(x, t) ≥∫

Ω

G(x, y, t− s)uk(y, s) dy, x ∈ Ω, t > s > 0,

we have u < ∞ a.e. in Ω × (0, T c), u(·, t) < ∞ a.e. in Ω for all t ∈ (0, T c), andu =∞ in Ω× (T c,∞).

Definition 27.1. We say that u blows up at t = Tmax(u0) completely if Tmax(u0)= T c(u0).

As we shall see below the notion of complete blow-up is different from the notionof global blow-up in Remark 24.6(v) and Section 43. In fact, the following theoremguarantees that the solution u from Theorem 24.1 (satisfying u(x, T ) ≤ Cα|x|−α)blows up completely.

Theorem 27.2. Consider problem (22.1) with p > 1, Ω bounded, 0 ≤ u0 ∈L∞(Ω), and Tmax(u0) <∞. Assume either

(i) p < pS

or

(ii) ut ≥ 0 in (0, Tmax(u0)).

Then u blows up completely at t = Tmax(u0).

See Remark 23.6 for conditions ensuring that ut ≥ 0. Theorem 27.2 is due to[54]. In Proposition 27.7 below, we shall see that the result may fail for p > pS .Before presenting the full proof of Theorem 27.2, we shall first give a proof of aspecial, one-dimensional case. This alternative approach is simpler than that inthe general case and, as an advantage, it can be used for problems with nonconvexnonlinearities. However, although the argument can be extended to dimensionsn > 1, the nonlinearity then has to satisfy severe growth restrictions and it requiresthe solution u to be increasing in time; see [54] for details.

Proof of Theorem 27.2 in a special case. We shall prove the assertion in case(ii), under the additional assumptions that n = 1, Ω = (−1, 1), and u0 is radialnonincreasing. These assumptions and Proposition 52.17 guarantee that ux ≤ 0for x ∈ [0, 1) and t ∈ (0, T ). Denote T := Tmax(u0).

Step 1. Denote f(u) = up. We shall prove ‖f(u(t))‖1 →∞ as t→ T−.Since u ≥ 0 and ut ≥ 0 we see that the function ψ : t → ‖f(u(t))‖1 is nonde-

creasing. Assume, by contrary, that ψ is bounded. Then the Lp-Lq-estimates andthe variation-of-constants formula guarantee

‖u(t)‖q ≤ ‖u0‖q +∫ t

0

(t− s)−α‖f(u(s))‖1 ds, α :=n

2

(1− 1

q

).

Since n = 1, in the particular case q =∞ we obtain

‖u(t)‖∞ ≤ ‖u0‖∞ + C

∫ t

0

(t− s)−1/2 ds < C(T ),

which contradicts T < ∞.Step 2. Denote ϕ(x) := limt→T− u(x, t) and let ε ∈ (0, 1). Then∫ 1−ε

−1+ε

f(ϕ(x)) dx = limt→T−

∫ 1−ε

−1+ε

f(u(x, t)) dx

≥ lim inft→T−

(1− ε)∫ 1

−1

f(u(x, t)) dx = ∞,

where we have used successively the monotone convergence of u to ϕ, ux ≤ 0 forx ≥ 0 and Step 1.

Step 3. Choose x ∈ (−1, 1), t > T . Then there exists ε > 0 such that t−T ≥ 2εand |x| < 1− ε. We have

(e−sAw)(x) =∫ 1

−1

G(x, y, s)w(y) dy ≥ Cε

∫ 1−ε

−1+ε

w(y) dy, s ∈ (ε, 2ε),

andCε := infG(x, y, s) : |x|, |y| < 1− ε, s ∈ (ε, 2ε) > 0.

Since e−tAu0 ≥ 0 and fk(uk(y, s)) ≥ fk(uk(y, T )) for s ≥ T , we obtain

uk(x, t) ≥∫ t

0

[e−(t−s)Afk(uk(s))

](x) ds

≥ Cε

∫ t−ε

t−2ε

∫ 1−ε

−1+ε

fk(uk(y, s)) dy ds ≥ Cε

∫ 1−ε

−1+ε

fk(uk(y, T )) dy.

(27.4)

Step 4. Using (27.4) and Step 2 we see that, as k →∞,

uk(x, t) ≥ Cε

∫ 1−ε

−1+ε

fk(uk(y, T )) dy → Cε

∫ 1−ε

−1+ε

f(ϕ(y)) dy =∞,

which proves the assertion.

An essential ingredient in the proof of Theorem 27.2 in the general case is thefollowing result (cf. [54, Lemma 2.1]), of independent interest, which is true for allp > 1 and without monotonicity assumption on u.

Proposition 27.3. Consider problem (22.1) with p > 1 and Ω bounded. Assumethat u0, v0 ∈ L∞(Ω) satisfy 0 ≤ v0 ≤ u0 and u0 ≡ v0. Then either Tmax(v0) =T c(u0) = ∞ or Tmax(v0) > T c(u0).

In [345], by employing arguments different from those in [54], complete blow-upwas proved for problem (19.1) for a rather general class of (convex nondecreas-ing) nonlinearities f(u), but only for monotone solutions in time (cf. case (ii) ofTheorem 27.2). Our proofs of Theorem 27.2 and Proposition 27.3 are based ona modification of ideas of [345] which enable one to cover case (i) as well. Letus also mention that the proof of Theorem 27.2(ii) in [54] can be used for moregeneral functions f only if either f satisfies serious growth restrictions or f is con-vex and the function f(u)/uγ is nondecreasing for large u, where γ > 1, see [54,Theorem 1].

In the proof of Proposition 27.3, we shall use the following two lemmas from[94]. The first one is an approximation lemma which will enable us to construct asuitable perturbation of the equation in (22.1).

Lemma 27.4. Let p > 1 and set ε0 := 1/(p+1). For each ε ∈ (0, ε0), there existsa concave function φε ∈ C2([0,∞)) with the following properties:

φε(0) = 0, 0 < φε(s) ≤ s for all s > 0, (27.5)

1 ≥ φ′ε(s) ≥ s−p

(φp

ε(s)− (p + 1)ε)+, s > 0, (27.6)

limε→0+

φ′ε(s) = 1 uniformly on [0, M ], for every M > 0, (27.7)

sups≥0

φε(s) < ∞. (27.8)

Proof. Let z = zε be the solution of the ODE

z′(s) = s−p(zp(s)− ε), s ≥ 1, with z(1) = 1− ε. (27.9)

We claim that z exists and satisfies

0 < z′(s) < 1, z(s) < s for all s ≥ 1. (27.10)

First observe that ε1/p < z(1) < 1, due to (1 − ε)p > 1 − pε > ε. The claim thuseasily follows from the fact that z(s)−s < 0 implies z′(s)−1 ≤ (z(s)/s)p−εs−p−1< 0, and that z(s) > ε1/p implies z′(s) > 0. Differentiating (27.9) and using(27.10), we get

z′′(s) = s−ppzp−1z′ − ps−p−1(zp − ε) = ps−p(zp−1 − sp−1)z′ ≤ 0, s ≥ 1.

Now extend z(s) to a concave function φε ∈ C2([0,∞)) verifying (27.5) and 0 ≤φ′

ε ≤ 1. (This is clearly possible since z′(1) < z(1) < 1.) We see that

φ′ε(s) ≥ φ′

ε(1) = (1− ε)p− ε > 1− (p + 1)ε ≥ s−p(φpε(s)− (p + 1)ε)+, 0 < s ≤ 1.

(27.11)This along with (27.9), (27.10) and 0 ≤ φ′

ε ≤ 1 proves (27.6). Next, by (27.9), wehave ∫ z(s)

z(1)

dτ

τp − ε=∫ s

1

dτ

τp< C :=

∫ ∞

1

dτ

τp<

∫ ∞

z(1)

dτ

τp − ε, s ≥ 1,

which yields (27.8). Finally, (27.7) is a consequence of the first inequality in (27.11),together with the continuous dependence of the solution zε of (27.9) on ε (observethat z(s) = s is solution of (27.9) for ε = 0).

Lemma 27.5. Assume Ω bounded and let 0 < T0 <∞. There exists K = K(T0) >0 such that the solution of the problem

Zt −∆Z = 1, x ∈ Ω, 0 < t ≤ T0,

Z = 0, x ∈ ∂Ω, 0 < t ≤ T0,

Z(x, 0) = −Kδ(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (27.12)

satisfies Z ≤ 0 in Ω× [0, T0].

Proof. Decompose Z as Z = Z1 − KZ2 where Z1 solves (27.12) with K = 0and Z2 = e−tAδ. In view of (1.4), we have Z2(x, t) ≥ c−1

2 e−λ1tϕ1 ≥ c1c−12 e−λ1tδ.

Combining this with Z1 ≤ c3(T0)δ (due to Z1 ∈ C1,0(Ω× [0, T0])) we obtain Z ≤(c3 −Kc1c

−12 e−λ1t)δ, and the lemma follows by choosing K = c−1

1 c2c3eλ1T0 .

Proof of Proposition 27.3. Assume for contradiction that Tmax(v0) ≤ T c(u0) <∞ or Tmax(v0) < ∞ = T c(u0). Let v be the solution of (22.1) with initial data v0.Notice that Tmax(u0) ≤ Tmax(v0) and fix τ ∈ (0, Tmax(u0)). By the assumptionson v0 and Proposition 52.7, there exists η > 0 such that

v(x, τ) + 2ηδ(x) ≤ u(x, τ), x ∈ Ω. (27.13)

Fix T ∈ (τ, Tmax(v0)).Step 1. L1- and Lp

δ-estimates. We claim that

u, upδ ∈ L1(QT ). (27.14)

Let uk be the solution of (27.1). Using Proposition 49.11 in Appendix C andT < T c(u0), we deduce that, for all small t > 0,

c(x, t)∫

Ω

uk(y, T )δ(y) dy ≤∫

Ω

G(x, y, t)uk(y, T ) dy

≤ uk(x, T + t) ≤ u(x, T + t) < ∞

for a.e. x ∈ Ω and some constant c(x, t) > 0. It follows that

supk

∫Ω

uk(y, T )δ(y) dy <∞. (27.15)

Now, for any 0 ≤ ϕ ∈ C2,1(Ω× [0, T ]) such that ϕ = 0 on ∂Ω× [0, T ], by testing(27.1) with ϕ we obtain∫

Ω

uk(y, T )ϕ(y, T ) dy =∫

Ω

u0(y)ϕ(y, 0) dy+∫ T

0

∫Ω

uk(ϕt+∆ϕ)+fk(uk)ϕ

dy ds.

First taking ϕ(x, t) = eλ1tϕ1(x) and using (1.4) and (27.15), we obtain

supk

∫ T

0

∫Ω

fk(uk)δ dy ds ≤ C supk

∫Ω

uk(y, T )δ(y) dy < ∞, (27.16)

hence upδ ∈ L1(QT ) by monotone convergence. Finally taking ϕ(x, t) = Θ(x),where Θ is defined by (19.27), and using (27.16), we similarly obtain u ∈ L1(QT ).

Step 2. Derivation of a penalized weak inequality for φε(u). Now fix ε ∈ (0, ε0)to be determined later and let φε be given by Lemma 27.4. For each k, a directcomputation yields

∂t(φε(uk))−∆(φε(uk)) = φ′ε(uk)(∂tuk −∆uk)− φ′′

ε (uk)|∇uk|2 ≥ φ′ε(uk)fk(uk)

in Ω× (τ, T ). For any 0 ≤ ϕ ∈ C2,1(Ω× [τ, T ]) such that ϕ = 0 on ∂Ω× [τ, T ] andϕ(T ) = 0, multiplying by ϕ and integrating by parts, it follows that∫

Ω

(φε(uk)ϕ

)(y, τ) dy +

∫ T

τ

∫Ω

φε(uk)(ϕt + ∆ϕ) + φ′

ε(uk)fk(uk)ϕ

dy ds ≤ 0.

(27.17)Set w := φε(u) and observe that w ∈ L∞(Ω × (0,∞)) by (27.8). Using (27.14),(27.5), (27.6) and passing to the limit in (27.17) via dominated convergence, weobtain ∫

Ω

(φε(u)ϕ)(y, τ) dy +∫ T

τ

∫Ω

w(ϕt + ∆ϕ) + φ′

ε(u)upϕ

dy ds ≤ 0.

For sufficiently small ε1 ∈ (0, ε0) and all ε ∈ (0, ε1], owing to (27.13) and (27.7),we have φε(u(·, τ)) ≥ v(·, τ) + ηδ(·). Using (27.6), it follows that∫

Ω

((v+ηδ)ϕ

)(y, τ) dy+

∫ T

τ

∫Ω

w(ϕt+∆ϕ)+(wp−(p+1)ε)+ϕ

dy ds ≤ 0. (27.18)

Step 3. Construction of a supersolution to the original problem and conclu-sion. Now let K and Z be given by Lemma 27.5 for T0 := Tmax(v0), selectε = minε1, η(K(p + 1))−1, and set

z(·, t) := w(·, t) + (p + 1)εZ(·, t− τ) ≤ w(·, t), τ < t < T0.

By (27.8), we have

supτ<s<T0

‖z(t)‖∞ ≤ M := sups≥0

φε(s) + (p + 1)ε sup0<s<T0

‖Z(s)‖∞ < ∞.

Combining (the weak formulation of) (27.12) with (27.18), it follows that∫Ω

(vϕ)(y, τ) dy +∫ T

τ

∫Ω

z(ϕt + ∆ϕ) + zpϕ

dy ds ≤ 0

for all ϕ as in Step 2. In other words, z is a bounded, weak supersolution toproblem (22.1) on [τ, Tmax(v0)), with initial data v(·, τ). By the weak comparisonprinciple (cf. Appendix F), it follows that v ≤ z ≤ M in Ω × (τ, Tmax(v0)): acontradiction.

Proof of Theorem 27.2 in case (i). By Theorem 22.13 and p < pS we knowthat the function

Tmax : L∞(Ω) → (0,∞] : u0 → Tmax(u0) (27.19)

is continuous. Fix u0 ≥ 0 with Tmax(u0) < ∞. Proposition 27.3 then implies

Tmax(u0) = limα→1−

Tmax(αu0) ≥ T c(u0).

In view of the proof in case (ii), we first make a simple observation.

Lemma 27.6. Let u0 ∈ L∞(Ω), u0 ≥ 0, and 0 < τ < Tmax(u0). Then T c(u(τ)) =T c(u0)− τ .

Proof. Let vk be the solution of (27.1) with u0 replaced by u(τ). For k large,we have fk(u(·, t)) = up(·, t) on [0, τ ], hence uk = u on [0, τ ] by uniqueness. Inparticular vk(0) = u(τ) = uk(τ), hence vk(t) = uk(t + τ) for all t ≥ 0, and theconclusion follows from the definition of T c.

Proof of Theorem 27.2 in case (ii). Fix τ ∈ (0, Tmax(u0)). Then u(τ) ≥ u0,and u(τ) ≡ u0 (since otherwise u would be a stationary solution, hence Tmax(u0) =∞). Proposition 27.3 and Lemma 27.6 then guarantee that

Tmax(u0) ≥ T c(u(τ)) = T c(u0)− τ.

Consequently, Tmax(u0) ≥ T c(u0).

We now present a different proof of Proposition 27.3, which is based on theoriginal ideas of [54] (cf. [54, Lemma 2.1]). This proof can be easily adapted toproblems with nonlinear boundary conditions (see [445]) and also to problems onunbounded domains. In the unbounded domain case, the modification of the proofbelow guarantees Tmax(αu0) > T c(u0) provided α < 1 and T c(u0) < ∞. Thisinformation is still sufficient for the proof of complete blow-up if we know that thefunction α → Tmax(αu0) is continuous (see the proof of Theorem 27.2 in case (i)).

Alternative proof of Proposition 27.3. Owing to Lemma 27.6, we may assumethat u0 ∈ C1(Ω), u0 > 0 in Ω, u0 = 0 and ∂u0/∂n < 0 on ∂Ω, and v0 := αu0

for some α ∈ (0, 1) (otherwise, just replace u0 by u(τ) for some small τ > 0 andobserve that v(τ) ≤ αu(τ) for some α ∈ (0, 1)). Let T ∈ (0,∞), T ≤ T c(u0), henceu(x, t) < ∞ a.e. in QT . We shall prove that there exists a constant Cα < ∞ suchthat u(x, t; αu0) ≤ Cα for all x ∈ Ω and t < T , which implies the conclusion (sincethen, Tmax(αu0) > T , hence Tmax(αu0) > T c(u0) if T c(u0) < ∞, Tmax(αu0) = ∞otherwise).

Let V := e−tAv0, and let ukλ, λ ∈ α, 1, k = 1, 2, . . . , be given by

∂tukλ −∆uk

λ = (uk−1λ )p in QT ,

ukλ = 0 on ST ,

ukλ(x, 0) = λu0(x), x ∈ Ω,

where u0λ :≡ 0. Notice that uk

λ ∈ C2,1(Ω× (0, T )) and that the maximum principleimplies

0 ≤ ukλ ≤ uk+1

λ ≤ u

ukα ≤ αuk

1

in QT . (27.20)

For m ∈ N, µ > 1, set

Emµ := (x, t) ∈ QT : um

α (x, t) > µV (x, t),

gmk (µ) := inf

(x,t)∈Emµ

uk1(x, t)

umα (x, t)

,

w(x, t) := uk+11 (x, t)− gm

k (µ)pumα (x, t) + µ

(gm

k (µ)p − gmk+1(µ)

)V (x, t).

(Here and below, we write gmk (µ)p in place of (gm

k (µ))p for simplicity.) Observethat Em

µ′ ⊂ Emµ for µ′ > µ, hence the functions gm

k are nondecreasing in µ. Set

M := supµ > 1 : Emµ = ∅ = infµ > 1 : Em

µ = ∅

and assume 1 < µ < M . Then w ∈ C2,1(Ω× (0, T )) and there exists δ = δ(m, µ) >0 such that t > δ for all (x, t) ∈ Em

µ . For k ≥ m > 1 we have

wt −∆w = (uk1)p −

(gm

k (µ)um−1α

)p,

w ≥ gmk+1(µ)um

α − gmk (µ)pum

α + µ(gm


)V in Em

µ ,

and, by (27.20),gm

k (µ) ≥ 1/α > 1, (27.21)

(uk1)

p ≥(gm

k (µ)umα

)p≥ (gmk (µ)um−1

α

)p in Emµ ,

hencewt −∆w ≥ 0 in Em

µ .

Since umα = µV on ∂Em

µ \(Ω× T

), we also have

w ≥ 0 on ∂Emµ \

(Ω× T

),

and we deduce from the maximum principle5 that w ≥ 0 in Emµ .

Assume that M > µ′ > µ > 1. We claim that

gmk+1(µ

′) ≥ gmk (µ)p −

(gm


) µ

µ′ . (27.22)

If gmk (µ)p − gm

k+1(µ) ≥ 0, then (27.22) follows by combining w ≥ 0 on Emµ ,

V (x, t) <1µ′ u

mα (x, t) for all (x, t) ∈ Em

µ′

and Emµ′ ⊂ Em

µ . If gmk (µ)p−gm

k+1(µ) < 0, then, using gmk+1(µ

′) ≥ gmk+1(µ), inequality

(27.22) reduces to µ′ ≥ µ.Now, the sequence gm

k (µ)k∈N is nondecreasing and bounded by infEmµ

u/umα <

∞. Its limit gm(µ) satisfies

gm(µ′) ≥ gm(µ)p −(gm(µ)p − gm(µ)

) µ

µ′ ,

5The set Emµ need not be connected nor cylindrical, but the corresponding maximum principle

can be proved by using similar arguments as in the proof of Proposition 52.4.

hence,gm(µ′)− gm(µ)

µ′ − µ≥ gm(µ)p − gm(µ)

µ′ . (27.23)

Fix µ0 ∈ (1, M), set

f(µ) := gm(µ0) +∫ µ

µ0

gm(s)p − gm(s)s

ds, µ ∈ [µ0, M),

and note that

f ′(µ) =gm(µ)p − gm(µ)

µa.e. in [µ0, M). (27.24)

As the function gm is nondecreasing, we know that its derivative exists a.e. andthat

gm(µ) ≥ gm(µ0) +∫ µ

µ0

(gm)′(ξ)dξ in [µ0, M). (27.25)

Since (gm)′ ≥ (gm(µ)p− gm(µ))/µ a.e. due to (27.23), it follows from (27.25) that

gm ≥ f in [µ0, M). (27.26)

Combining (27.24), (27.26) and (27.21), we infer f ′(µ) ≥ (f(µ)p − f(µ))/µ a.e.Integrating this inequality and using (27.21) again we obtain

log(µ/µ0) ≤∫ ∞

gm(µ0)

dσ

σp − σ≤∫ ∞

1/α

( σp−2

σp−1 − 1− 1

σ

)dσ = log

((1− αp−1)−1/(p−1)

)for all 1 < µ0 < µ < M . Consequently,

M ≤ cα := (1 − αp−1)−1/(p−1).

Since Emµ = ∅ for µ > M , we have

umα (x, t) ≤ cαV (x, t) in QT .

Since the limit Uα(x, t) := limm→∞ umα (x, t) is a bounded integral solution of (22.1)

with u0 replaced by αu0, it coincides with u(x, t; αu0) for t < T . This concludesthe proof.

The following result shows that incomplete blow-up may occur when p > pS .Parts (i) and (ii) are respectively due to [396] and [232].

Proposition 27.7. Consider problem (22.1) with Ω bounded and p > 1. Let 0 ≤ϕ ∈ L∞(Ω), ϕ ≡ 0, let α∗ be defined by (22.22) and set u0 = α∗ϕ.(i) Then T c(u0) = ∞.(ii) Assume in addition Ω = BR, ϕ radial and p > pS. Then Tmax(u0) < ∞.Consequently, u blows up incompletely as t = Tmax.

Proof. (i) Let 0 ≤ α < α∗. As a consequence of the definition of α∗ and of thecomparison principle, we have Tmax(αϕ) = ∞. Let vα be the solution of (22.1)with initial data αϕ, and let vα,k and uk be the (global) solutions of (27.1), withinitial data αϕ and u0 respectively. Since Tmax(αϕ) =∞, Theorem 17.1 implies∫

Ω

vα(t)ϕ1 dx ≤ C = C(Ω, p), for all t > 0.

Since vα,k ≤ vα, it follows that∫Ω

vα,k(t)ϕ1 dx ≤ C, hence∫Ω

uk(t)ϕ1 dx ≤ C, for all t > 0,

by continuous dependence. Letting k → ∞ and using monotone convergence, wededuce that, for each t > 0,

∫Ω

u(t)ϕ1 dx ≤ C, hence u(x, t) < ∞ for a.e. x ∈ Ω.We conclude that T c(u0) =∞.

(ii) This assertion is a consequence of Theorem 28.7 below.

Remarks 27.8. (a) If Ω is a ball and u0 is radial, then the assumption p < pS

in Theorem 27.2 can be weakened to p ≤ pS (or can be removed if we knowB(u0) = 0), see [232, the proof of Theorem 5.1].

(b) For some class of problems (including (22.1)), property (22.28) is sufficientfor complete blow-up, see [54, Corollary 3.1].

(c) Genericity of complete blow-up. Consider the situation in Proposi-tion 27.7. If u0 = αϕ, α > α∗, then T c(u0) < ∞. More precisely, Tmax(αϕ) ≤T c(αϕ) < Tmax(βϕ) for any α > β ≥ α∗. This follows from Proposition 27.3 (seealso [317, Theorem 2] and [232, Theorem 14.1]). Since the function

[α∗,∞)→ (0,∞) : α → Tmax(αϕ) (27.27)

is decreasing, it has to be (left) continuous a.e., hence T c(αϕ) = Tmax(αϕ) fora.e. α > α∗. Note that if we were able to prove the blow-up rate (23.5) for u0 =αϕ, α ≥ α∗, with M = M(α) being locally bounded, then the proof of [254,Theorem 1.2] would guarantee the continuity of (27.27) everywhere. Recall alsothat (23.5) is true if pS ≤ p < pJL, cf. Theorem 23.10.

(d) Peaking solutions. The facts mentioned in (c) show that solutions whichblow up incompletely are rather exceptional. Many interesting results on the be-havior of such solutions in the interval (Tmax(u0), T c(u0)) can be found in [194],[379], [380], [382], [195].

In [232] the authors considered problem (18.1) with pS T,

where f = f(ρ) and g = g(ρ) are suitable bounded positive solutions of the ODE’s

f ′′ +n− 1

ρf ′ − 1

2f ′ρ− 1

p− 1f + fp = 0, ρ > 0, f ′(0) = 0,

andg′′ +

n− 1ρ

g′ +12g′ρ +

1p− 1

g + gp = 0, ρ > 0, g′(0) = 0,

respectively. These solutions have singularity (peak) only at the point (0, T ) andtheir blow-up profile is limt→T u(r, t) = Cr−2/(p−1) for some C < cp. Similarsolutions for problem (22.21) had been previously constructed in [318].

(e) Incomplete blow-up in the subcritical case. Another explicit exampleof incomplete blow-up, for the nonautonomous equation

ut −∆u = a(|x|, t)u2, x ∈ Rn, t ∈ R, (27.28)

(with a > 0 being bounded above and bounded away from zero), is due to [414].Let ϕ ∈ BC1(R) be nonnegative. Set

u(x, t) :=1

ϕ(t) + r2, where r = |x|.

Then a straightforward computation yields

ut −∆u = a(|x|, t)u2 if x = 0 or ϕ(t) = 0,

where

a(r, t) := 2n− ϕ′(t)− 8r2

ϕ(t) + r2

is bounded above anda ≥ 2(n− 4)− sup ϕ′ > 0

provided n > 4 and supϕ′ is small enough. In addition, it is easily verified that uis a weak solution of (27.28) if n > 4. Notice that p = 2 is subcritical if n = 5.

In particular, if n > 4 and ϕ(t) = t2 for |t| < n− 4, then ∞ > C2 ≥ a(|x|, t) ≥C1 > 0 in R

n × (−1/2, 1/2) and the solution u exhibits incomplete blow-up att = 0. Similarly, the choices ϕ(t) = [t(1− t)]2 or ϕ(t) = t3(sin 1

t )2 yield examples of

functions u which blow up incompletely multiple or infinitely many times (cf. [380],[382] in the case a ≡ 1).

Using the example above one can easily construct explicit examples of incom-plete blow-up for the problem

ut −∆u = a(x, t)u2 + b(x, t), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

where Ω ⊂ Rn is bounded, n > 4, a, b are bounded and a > 0 is bounded away from

zero, see [442]. On the other hand, it was shown in [442] that incomplete blow-upcannot occur in such problems in the subcritical range p < pS if a, b ∈ BUC1.

(f) Analytic continuation. A different possible way of continuing the solutionafter Tmax was studied in [351]. It is based on a suitable notion of analytic contin-uation where the time variable t is extended to a sector in the complex plane. Theexistence of such continuation was proved there for the equation ut − ∆u = u2,under Neumann boundary conditions and suitable assumptions on the initial data.

(g) Critical Lq-space. Like the classical existence time, the complete blow-uptime T c(u0) is not uniformly positive for bounded sets of initial data in Lq(Ω)when q = qc = n(p− 1)/2 (cf. [36]). Indeed, assume for instance Ω bounded, Ω ⊃B(0, 1), and let u0,j = 2u0,j, where u0,j is given by (15.3). Then Proposition 27.3and Remark 15.4(i) imply that T c(u0,j) < Tmax(u0,j) → 0 as j → ∞, while‖u0,j‖qc = Const.

28. Applications of a priori bounds

We have seen in previous sections that a priori and universal estimates of solu-tions play a key role in the proofs of several important statements. In this sectionwe provide further applications of such estimates. Other applications (concerningexistence of nodal equilibria and connecting orbits) can be found in [128] and [3],for example. These articles are devoted to superlinear problems with nonlinearboundary conditions and indefinite nonlinearities, respectively.

28.1. A nonuniqueness result

In this subsection we use universal bounds from Section 26 and arguments of[53] in order to prove Theorem 15.3(ii). More precisely, we prove the followingproposition.

Proposition 28.1. Let Ω = BR and pF qc = n(p − 1)/2and assume that u0 ∈ Lr(Ω), u0 ≥ 0, is radial nonincreasing. Let Tm denote themaximal existence time of the corresponding classical Lr(Ω)-solution um of (15.1)(cf. Theorem 15.2 and Proposition 16.1) and let T ∈ (0, Tm). Then there exists a

28. Applications of a priori bounds 231

function u ≥ um, u = um, such that u is a classical Lq(Ω)-solution of (15.1) forany q ∈ [1, qc), u(·, t) is radial nonincreasing,

limt→0

‖u(·, t)‖q =∞ for any q > qc, (28.1)

limt→T

‖u(·, t)‖q = ∞ for any q > qc. (28.2)

In the proof of Proposition 28.1 we will also need the following lemma.

Lemma 28.2. Let Ω = BR, p > pF , 0 < T < ∞ and let u be a positive, radialnonincreasing classical solution of (15.1) in the time interval (0, T ). Let δ > 0and T ′ ∈ (0, T ). Then there exist constants c depending only on R, p, n and theindicated quantities, such that

u(x, t) ≤ c(δ)|x|−2/(p−1), |x| ≤ R, t ∈ (0, T − δ], (28.3)

‖u(·, t)‖q ≤ c(q, δ), t ∈ (0, T − δ], 1 ≤ q < qc, (28.4)∫ T ′

0

‖u(·, s)‖pp ds < c(T ′, T − T ′). (28.5)

Proof. Let t ∈ (0, T − δ]. Denote β = 1/(p − 1) and v := u(·, t). Then (15.22)guarantees

‖sβe−sAv‖∞ ≤ Cp, for all s ∈ (0, δ]. (28.6)

Let us show the existence of constants C, k > 0 such that

v(x) ≤ Cekδ/R2(R2/δ)β |x|−2β , |x| ≤ R. (28.7)

If k > 0 is sufficiently large, then there exists η ∈ D(B1) radial, radially decreasing,η ≡ 0, such that −∆η ≤ kη (one can take η(x) = exp

[−1/(1 − 2|x|2)+

], for

example). Set ηλ(x) := η(λx), λ ≥ 1/R. Then the support of ηλ is a subset of Ωand

−∆ηλ ≤ kλ2ηλ,

hencee−sAηλ ≥ e−kλ2sηλ

by the maximum principle. Consequently, (28.6) guarantees

Cp

∫Ω

ηλ dx ≥∫

Ω

sβ(e−sAv)ηλ dx = sβ

∫Ω

v(e−sAηλ) dx ≥ sβe−kλ2s

∫Ω

vηλ dx.

Since v(x) ≥ v(1/λ) on the support of ηλ, we obtain

v( 1

λ

)≤ Cps

−βekλ2s, λ ≥ 1R

, s ∈ [0, δ].

Choosing λ = 1/|x| and s = δ|x|2/R2 we obtain (28.7).

Notice that (28.7) guarantees (28.3) and (28.4) is a consequence of (28.3). Forτ ∈ (0, t), multiplying the variation-of-constants formula between τ and t by ηλ

we obtain∫Ω

(e−(t−τ)Au(τ))ηλ dx +∫ t

τ

∫Ω

up(s)(e−(t−s)Aηλ) dx ds =∫

Ω

u(t)ηλ dx.

It follows that ∫ t

0

∫Ω

up(s)e−kλ2(t−s)ηλ dx ds ≤∫

Ω

u(t)ηλ dx.

Fixing λ0 ≥ 1/R and r0 > 0 such that ηλ0(r0) > 0 and using (28.3) we obtain

ηλ0(r0)e−kλ20T ′∫ t

0

∫Br0

up(s) dx ds ≤∫

Ω

u(t)ηλ0 dx ≤ C(T − T ′), t ≤ T ′ < T.

Since u(·, t) is radial decreasing, the last estimate guarantees (28.5).

Proof of Proposition 28.1. Fix T ∈ (0, Tm). Let ηk ∈ D(B1/k), k = 1, 2, . . . , benonnegative, radial, radially decreasing and ηk ≡ 0. Fix k. Due to the continuousdependence on initial data (see Remark 51.8(iii)) we have Tmax(u0 + αηk) > Tfor α > 0 small. On the other hand, as a consequence of Remark 17.2(i) (see alsoRemark 17.7(iv)), we have Tmax(u0 +αηk) < T for α > 0 large. Since the mappingα → Tmax(u0 + αηk) is continuous (see Theorem 22.13 and (51.92)) there existsαk > 0 such that Tmax(u0 + αkηk) = T .

Let uk denote the Lr(Ω)-solution of (15.1) with initial data u0 + αkηk. Due toTheorem 26.8 the sequence uk is uniformly bounded on Ω × [δ, T − δ] for anyδ > 0. Now parabolic regularity estimates (see Theorems 48.1 and 48.2) imply auniform bound in BUC2+α,1+α/2(Ω × [δ, T − δ]) for some α > 0, hence we mayassume uk → u in C2,1(Ω× [δ, T −δ]) for all δ > 0, where u is a classical solution of(26.2). Passing to the limit in the variation-of-constants formula for uk we obtain

u(t) = e−(t−s)Au(s) +∫ t

s

e−(t−σ)Aup(σ) dσ. 0 < s < t < T. (28.8)

Moreover, applying Lemma 28.2 to the uk’s, and then Fatou’s lemma, we deducethat u satisfies (28.3)–(28.5). Next fix q ∈ [1, qc) and let t ∈ (0, T ). Inequality(28.4) and the compactness of e−tA show that there exist sm → 0 and w ∈ Lq(Ω)such that

u(sm) → w weakly in Lq(Ω) and

e−(t−sm)Au(sm) → e−tAw in Lq(Ω).

Since (28.5) guarantees up ∈ L1(QT ′) for all T ′ < T , using (28.8) with s = sm andpassing to the limit we obtain

u(t) = e−tAw +∫ t

0

e−(t−σ)Aup(σ) dσ, 0 < t < T. (28.9)

Next we show w = u0 a.e. Let φ ∈ D(Ω). Multiplying the equation for uk with φand integrating we obtain∫

Ω

uk(t)φdx−∫

Ω

(u0+αkηk)φdx+∫ t

0

∫Ω

uk(s)(−∆φ) dx ds =∫ t

0

∫Ω

uk(s)pφdx ds.

Assume φ ≡ 0 on Bε for some ε > 0. Since uk ≤ C(ε) on (Ω\Bε)× (0, T ), we maypass to the limit via dominated convergence in the above identity, and we arriveat∫

Ω

u(t)φdx −∫

Ω

u0φdx +∫ t

0

∫Ω

u(s)(−∆φ) dx ds =∫ t

0

∫Ω

up(s)φdx ds. (28.10)

On the other hand, (26.2) shows that∫Ω

u(t)φdx−∫

Ω

u(sm)φdx +∫ t

sm

∫Ω

u(s)(−∆φ) dx ds =∫ t

sm

∫Ω

up(s)φdx ds.

(28.11)Passing to the limit in (28.11) and comparing the resulting identity with (28.10)yields

∫Ω

u0φdx =∫Ω

wφdx for all φ ∈ D(Ω) which vanish in a neighborhood ofthe origin, hence u0 = w a.e.

Now (28.9) guarantees ‖u0 − u(t)‖1 → 0 as t → 0. This convergence, (28.4)and interpolation yield ‖u0 − u(t)‖q → 0 as t → 0 for any q < qc, hence u is anLq(Ω)-solution of (15.1) for any q < qc. It remains to prove (28.1) and (28.2). Fixq > qc. We know that uk(T/2)→ u(T/2) in Lq(Ω). Due to the continuity of Tmax

(cf. above) we have

Tmax(u(T/2)) = limk→∞

Tmax(uk(T/2)) = T/2,

hence ‖u(t)‖q → ∞ as t → T due to Remarks 16.2. Next assume that there existC > 0 and tk → 0 such that ‖u(tk)‖q < C. Choose q ∈ (qc, q). Then interpolationyields u(tk) → u0 in Lq(Ω) and the continuity of Tmax in Lq(Ω) shows

T = limk→∞

Tmax(u(tk)) = Tmax(u0) = Tm > T,

a contradiction. This shows (28.2) and concludes the proof.

Remark 28.3. Entire solutions. Let us mention another, simple application ofthe universal bounds in Section 26. Let Ω = BR, 1 < p < pS , and denote by φ theunique positive solution of (6.1) with λ = 0 (cf. Remark 6.9(ii)); we know that φ isradial. One can show that any entire, radial, positive classical solution u of (15.1)(i.e. defined for all t ∈ R) is either φ or a connection from φ to 0. Moreover, thisremains true without the assumption that u be radial if we assume 1 < p < pB.

Indeed, due to Theorem 26.8, any such solution satisfies supt∈R ‖u(t)‖∞ < ∞,hence supt∈R

‖u(t)‖BUC1+α(Ω) < ∞ by smoothing effects. Owing to the (strict)Lyapunov functional given by the energy E (cf. (17.6)), we know from Proposi-tion 53.5 that the ω-limit set of u (in the BUC1-topology) is nonempty and consistsof nonnegative equilibria. By the same token, this is also true for the α-limit set(obtained by taking tk → −∞ instead of +∞ in formula (53.1)). Now, using thefact that φ and 0 are the only nonnegative steady states, that E′(t) ≤ 0, and thatE(φ) > E(0) = 0, one easily obtains the conclusion.

28.2. Existence of periodic solutions

Analogously as in Corollary 10.3, a priori estimates for positive periodic solutionsof (suitable) parabolic problems with periodic superlinear nonlinearities guaranteetheir existence. For example, consider the problem

ut −∆u = f(x, t, u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0.

(28.12)

If Ω = BR, f = f(|x|, t, u) is continuous, T -periodic in t, 1 0, u ≥ 0,

and, for all (x, t) in the closure of Q := QT ,

limu→∞, Q(z,τ)→(x,t)

u−pf(z, τ, u) = m(x, t) ∈ (0,∞),

then a straightforward generalization of Theorem 26.8 shows that any positive T -periodic solution of (28.12) is bounded by a universal constant C = C(f, Ω) (see[425] for more general statements). Consequently, if f satisfies additional assump-tions guaranteeing the well-posedness of (28.12) in a suitable function space, thena topological degree argument shows the existence of a positive T -periodic solutionof (28.12) (see [176] for details concerning the use of the topological degree).

Of course, instead of the radial symmetry assumption we could have assumedp < pB. Let us sketch another proof of universal estimates of positive T -periodicsolutions of (28.12) in the general nonradial case and the full subcritical range 1 <p < pS . Unfortunately, this alternative proof requires quite restrictive assumptionsconcerning the nonlinearity f .

Proposition 28.4. Assume Ω bounded and f(x, t, u) = m(t)|u|p−1u, where 1 0

m′(t)−m(t)

<2n− (n− 2)(p + 1)

r2(Ω), (28.13)

where r(Ω) denotes the radius of the smallest ball containing Ω. Then there existsa constant C > 0 such that any positive T -periodic solution of (28.12) satisfies

‖u(t)‖∞ ≤ C for all t > 0. (28.14)

Consequently, there exists at least one positive T -periodic solution of (28.12).

Sketch of proof. Let u be a positive T -periodic solution of (28.12). Multiplyingby ϕ1 one easily gets

ψ′ ≥ −λ1ψ +[inft>0

m(t)]ψp, ψ(t) :=

∫Ω

u(t)ϕ1 dx,

hence∫Ω u(t)ϕ1 dx ≤ C. If Ω is convex, then the method of moving planes guar-

antees u(x, t), |∇u(x, t)| ≤ C for all x in a neighborhood of ∂Ω. If Ω is not convex,then the same estimate can be obtained by using the Kelvin transform (cf. theproof of Theorem 13.1). Now, the Pohozaev-type identity∫ T

0

∫Ω

[ n

p + 1− n− 2

2

]up+1m(t) dx dt =

∫ T

0

∫Ω

[m′(t)p + 1

up+1 + u2t

] |x|22

dx dt

+12

∫ T

0

∫∂Ω

|∇u|2(x · ν(x)

)dσ dt,

the identity ∫ T

0

∫Ω

u2t dx dt = − 1

p + 1

∫ T

0

∫Ω

m′up+1 dx dt

and the assumption (28.13) guarantee an a priori bound for u in W 1,2(QT

). Finally

it is sufficient to use the bootstrap procedure from the proof of Theorem 22.1.

Remarks 28.5. (i) The estimates in Proposition 28.4 were first proved in [176]and [177] under the additional assumptions p(3n− 4) < 3n + 8 and p(n− 2) < n,respectively. The general case was proved in [441], cf. also [286]. Analogous resultsfor f(x, t, u) = |u|p−1u + h(x, t), h “small”, can be found in [280].

(ii) If Ω, f are as in Proposition 28.4 and we consider problem (28.12) comple-mented with the initial condition u(x, 0) = u0(x), then the a priori bound (22.27)is true for all solutions of (28.12) (not necessarily positive or periodic) and evenwithout assuming (28.13), see [441, Theorem 5.1(i)].

28.3. Existence of optimal controls

The a priori estimate (22.27) also plays an important role in the proof of existenceof optimal controls for problems with final observation. Let Ω ⊂ R

n be bounded,T > 0, 1 < p < pS , q ≥ 2, ud ∈ Lq(Ω), u0 ∈ C2(Ω) ∩ C0(Ω), and let us considerthe model optimal control problem

Minimize J(u(w), w) over w ∈ L2(Ω), (28.15)where

J(u, w) =∫

Ω

|u(x, T )− ud(x)|q dx +∫

Ω

w2 dx,

and u(w) is the solution of the governing equationut −∆u = |u|p−1u + w, x ∈ Ω, t ∈ (0, T ],

u = 0, x ∈ ∂Ω, t ∈ (0, T ],

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (28.16)

(we set J(u(w), w) := ∞ if (28.16) does not possess global solution up to time T ).Then we have the following

Proposition 28.6. Under the above assumptions, let

q ∈(n

2(p− 1),

2n

(n− 4)+

).

Assume that problem (28.16) possesses a global solution at least for one w ∈ L2(Ω).Then the optimal control problem (28.15) has a solution.

The statement of Proposition 28.6 remains true for more general time-dependentcontrols w ∈ Lr([0, T ], L2(Ω)) (where r is large enough) and more general costfunctionals J , see [22]. In addition, one can also derive optimality conditions foroptimal controls and show that the assumption p < pS is essentially optimal (see[22]).

Sketch of proof of Proposition 28.6. Let wk ⊂ L2(Ω) be a minimizingsequence for J and uk := u(wk). Then wk is bounded in L2(Ω) (and we mayassume wk → w weakly in L2(Ω)) and uk(T ) is bounded in Lq(Ω), due to theboundedness of J(uk, wk). Since the problem (28.16) is well-posed in Lq(Ω) we mayfind δ > 0 such that the solutions uk can be continued on the interval [T, T + δ].A straightforward modification of the proof of estimate (22.27) shows that thesolutions uk are uniformly bounded in L∞((0, T ), L2p(Ω)). The Sobolev maximalregularity (see Theorem 51.1(vi)) guarantees that uk are uniformly bounded inW 1,r([0, T ], L2(Ω)) ∩ Lr([0, T ], W 2,2 ∩W 1,2

0 (Ω)) for any r > 1. Since this space iscompactly embedded in X := C([0, T ], W 1,2

0 ∩ Lq(Ω)) for r sufficiently large (seeProposition 51.3), we may assume uk → u in X . Now it is easy to pass to the limitto show u = u(w) and J(u, w) ≤ limk→∞ J(uk, wk).

28.4. Transition from global existence to blow-up andstationary solutions

Let us consider problem (22.1) with either Ω bounded and p > 1, or Ω = Rn and

p > pF , and let us go back to the situation introduced in Subsection 22.3. Namely,fix a function φ ≥ 0, φ ≡ 0, with φ ∈ L∞(Ω) if Ω is bounded and, for instance,φ ∈ D(Ω) if Ω = R

n. Let α∗ be again defined by

α∗ = α∗(φ) := supα > 0 : Tmax(αφ) = ∞, (28.17)

and note that α∗ ∈ (0,∞) (cf. Subsection 22.3 for Ω bounded; when Ω = Rn this

follows from similar arguments by using Theorem 20.1). By definition of α∗, wehave

T (αφ) < ∞ for α > α∗

and, as a consequence of the comparison principle,

T (αφ) =∞ for 0 ≤ α < α∗.

Now if we consider the threshold solution u∗ := u(t; α∗φ) of (22.1) starting atu0 = α∗φ, we have the following three possibilities for u∗:

(a) u∗ is global and bounded in L∞(Ω),(b) u∗ is global but unbounded,(c) u∗ blows up in finite time.

It turns out that any of these three possibilities may occur.

Theorem 28.7. Consider the situation described above.(i) Assume either 1 pS. Assume either Ω = BR and φ radial, or Ω = R

n and φ radialnonincreasing. Then case (c) occurs.(iv) If Ω is bounded and case (a) occurs, then the ω-limit set of the solution u∗ is anonempty compact connected set consisting of positive equilibria. As a consequence,if Ω is a bounded starshaped domain and p ≥ pS, then (b) or (c) occurs, and thea priori bound (22.2) fails.

Proof. First let us show that the bound (22.2) guarantees alternative (a). Forany α ∈ (0, α∗), the solution uα(t) := u(t; αφ) exist globally. If (22.2) is true, then‖uα(t)‖∞ ≤ C∗ for some C∗ independent of α and the continuous dependence ofthe solutions on the initial data shows ‖u∗(t)‖∞ ≤ C∗, hence case (a) occurs. Since(22.2) is true if p < pS and either Ω is bounded or Ω = R

n and u0 is radial (seeTheorem 22.1 or Theorem 26.9, respectively), we have (a) in these cases.

If (a) is true and Ω is bounded, then Example 53.7 guarantees that the ω-limitset ω(α∗φ) consists of positive equilibria. Since (22.1) does not possess positiveequilibria if Ω is starshaped and p ≥ pS (see Corollary 5.2), the alternative (a)and hence the estimate (22.2) cannot be true in this case.

Now assume p = pS , Ω = BR and φ radial. If (c) occurred, then u wouldblow up completely at t = Tmax(u0), due to [232, the proof of Theorem 5.1]. Butthis would contradict Proposition 27.7(i). Consequently, (b) is the only remainingpossibility and assertion (ii) is proved. In the case of radial nonincreasing functionsthe assertion follows from Theorem 22.9.

Finally, let p > pS . If Ω is a ball and φ is radially symmetric, then again(a) cannot happen, and (b) is ruled out by Theorem 22.4. Consequently, (c) istrue. If Ω = R

n and φ is radial nonincreasing, then the result follows from [374,Theorem 1.1]) provided p < pL. If p ≥ pL, then one can use [374, Lemma 3.2] and[378].

Remarks 28.8. (i) Non-threshold solutions. Assume that either p < pS andΩ is bounded, or p > pS , Ω = BR and φ is radial. Then limt→∞ ‖u(t; αφ)‖∞ = 0for all 0 ≤ α < α∗. This follows from Proposition 19.11 and the boundedness ofglobal solutions (cf. Theorems 22.1 and 22.4).

(ii) Dynamical proofs of existence of equilibria. Let Ω be bounded andp < pS . Then similarly as above, ω(α∗φ) consists of nontrivial equilibria for any(possibly sign-changing) φ ∈ Lq(Ω) \ 0, q > qc, and this fact (together with atopological degree argument) can be used for the proof of existence of positive andsign-changing stationary solutions of (22.1) and related problems (see [114], [436],[439], [441], [3]).

(iii) Threshold solutions in the nonradial supercritical case. Assumethat Ω is bounded and convex, p > pS, and φ ∈ L∞(Ω) is nonnegative, φ ≡ 0. Letα∗ = α∗(φ) and u∗ have the same meaning as above. Fix αk α∗ and denote

u(t) := limk→∞

u(t; αkφ).

Then estimates in [396] show that u is a global weak solution of (22.1) and u(t) =u∗(t) for t ∈ [0, Tmax(α∗φ)) (cf. also Section 27). Recent results in [132] guaranteethat Tmax(α∗φ) < ∞ and there exists a compact set S ⊂ Ω× [Tmax(α∗φ),∞) suchthat the Hausdorff measureHn−4/(p−1)(S) is zero, u is continuous in Ω×(0,∞)\S,limt→∞ ‖u(t)‖∞ = 0. In particular, Theorem 28.7(iii) remains true in the nonradialcase if Ω is bounded and convex.

(iv) Further results and references on threshold solutions can be found in thefollowing subsection, Remarks 27.8 and Section 29.

28.5. Decay of the threshold solution of the Cauchy problem

In this subsection we denoteβ :=

1p− 1

,

and the notation f(t) ∼ g(t) for t ≥ t0 means that

C1g(t) ≤ f(t) ≤ C2g(t) for all t ≥ t0 and some constants C1, C2 > 0.

Consider the Cauchy problem (18.1) with p > pF . We continue to study thesituation described at the beginning of the previous subsection. In what follows,by non-threshold solutions we more specifically mean solutions correspondingto α ∈ (0, α∗).

Let us first consider the case of initial data with exponential spatial decay, moreprecisely φ ∈ H1

g , and assume also p < pS . Recall from Proposition 20.13 that if uis global and t0 > 0, then there exists k ≥ 0 such that

‖u(t)‖∞ ∼ t−λLk , t ≥ t0, (28.18)

where λLk = (n + k − 1)/2 for k ≥ 1 and λL

0 = β. The following theorem is due to[301].

Theorem 28.9. Let pF 0, denote by

uα the solution of (18.1) with initial data u0 = αφ and let α∗ be defined by (28.17).Then α∗ ∈ (0,∞). Moreover:

(a) uα is global and ‖uα(t)‖∞ ∼ t−n/2 for t ≥ 1 if 0 < α < α∗;(b) uα∗ is global and ‖uα∗(t)‖∞ ∼ t−β for t ≥ 1;(c) uα blows up in finite time if α > α∗.

Proof. Assertion (c) follows from the definition of α∗.Let vα denote the rescaled solution (see (18.13)). The asymptotic stability of the

zero equilibrium of (18.14) (see Example 51.24) shows that vα is global, vα(s)→ 0in H1

g (and L∞) if α is small and vα∗(s) → 0 in H1g as s → ∞. In particular,

α∗ > 0.If φ is radial, then Theorem 26.9 guarantees that uα∗ (hence vα∗) are global.

In the general case one can use the estimates in [301] or [478, Theorem 1] (seealso [440, Theorem 1.2] in the case of sign-changing solutions). The arguments inthe proof of Proposition 20.13 show C1 ≤ ‖vα∗(s)‖∞ ≤ C2, hence (28.18) is truewith k = 0. In addition, the compactness of the semiflow for problem (18.14),the existence of the Lyapunov functional and the stability of the zero equilibriumguarantee that the ω-limit set ω(vα∗) of vα∗ in H1

g is nonempty and consists ofpositive equilibria (cf. Theorem 28.7). For further reference fix w∗ ∈ ω(vα∗) and asequence sj →∞ such that vα∗(sj)→ w∗.

Fix α < α∗ and assume that vα(s) → 0. Then the arguments above show thatthere exists a subsequence of vα(sj) which converges to a positive equilibrium w.Now the proof of Theorem 19.9(ii) guarantees that vα(s) ≤ (α/α∗)vα∗(s), hencew < w∗. However, the proof of Proposition 19.8 shows that (18.14) does not possessordered positive equilibria. Consequently, vα(s) → 0 as s → ∞. Now the upperbound in (28.18) with k = 1 follows from Example 51.24 and the lower boundfrom the comparison with the solution of the linear problem (cf. (20.5)).

Theorem 28.9 shows that for positive φ ∈ H1g and p < pS, the threshold solution

decays with the self-similar rate t−β while the non-threshold solutions decay withthe same rate as the corresponding solutions of the linear heat equation. Thenext theorem [443] and subsequent remarks show that the same behavior of non-threshold solutions can be expected in a more general case, while the behavior ofthe threshold solution strongly depends on the exponent p.

Theorem 28.10. Assume p > pF . Let φ ∈ C(R+) be nonnegative, φ ≡ 0, and

limr→∞φ(r)r2β = 0. (28.19)

Denote by uα the solution of (18.1) with u0(x) = αφ(|x|), α > 0, and let α∗ bedefined by (28.17). Then α∗ ∈ (0,∞) and the following assertions are true.

(i) Let p < pS. Then uα∗ is global and

‖uα∗(t)‖∞ ∼ t−β , t ≥ 1. (28.20)

If α ∈ (0, α∗), thenlim

t→∞ ‖uα(t)‖∞tβ = 0. (28.21)

(ii) Let p ≥ pS. If uα∗ is global, then

lim supt→∞

‖uα∗(t)‖∞tβ = ∞. (28.22)

If α ∈ (0, α∗) and ‖uα(t)‖∞ ≤ ct−β for all t > 0, then (28.21) is true.

Remarks 28.11. (i) If limr→∞ φ(r)r2β =∞, then uα blows up in finite time forany α > 0 due to Theorem 17.12. If

0 < lim infr→∞ φ(r)r2β ≤ lim sup

r→∞φ(r)r2β <∞ (28.23)

and p < pS , then (28.20) remains true. In fact, the proof of Theorem 28.10 showsthat the threshold solution uα∗ satisfies the upper bound in (28.20). The lowerbound ‖uα∗(t)‖ ≥ ct−β follows from the comparison with the solution of the linear

problem and Lemma 20.8. Similarly, if p ≥ pJL, then one can replace condition(28.19) in the proof of (28.22) with the condition

lim supr→∞

α∗φ(r)r2β < cp, (28.24)

where cp is the constant from (3.9), see [443].(ii) Assume that

lim supr→∞

φ(r)r2η <∞ for some η > β.

Then estimate (28.21) guarantees that the solution behaves like the solution of thelinear problem. In fact, set h(t) := ‖uα(t)‖∞ and notice that the function w(t) :=exp[−

∫ t

0 h(s)p−1 ds]uα(t) is a subsolution of the linear heat equation (cf. [507,Proposition 2.6]). Assuming η ∈ (β, n) without loss of generality, Lemma 20.8thus implies

h(t) = exp[∫ t

0

h(s)p−1 ds]‖w(t)‖∞ ≤ Ct−η exp

[∫ t

0

h(s)p−1 ds], t > 1,

and (28.21) guarantees h(t)tβ → 0 as t → ∞. Choose ε > 0 such that κ :=η − εp−1 > β and fix t0 > 1 such that h(t) ≤ εt−β for t ≥ t0. Let t ≥ t0. Then∫ t

0

h(s)p−1 ds ≤∫ t0

0

h(s)p−1 ds + εp−1 log( t

t0

)=: I0 + εp−1 log

( t

t0

),

hence h(t) ≤ Ct−ηeI0(t/t0)εp−1= C0t

−κ, thus H :=∫∞0

hp−1(t) dt < ∞. Now wesee that

e−tA(αφ) ≤ uα(t) ≤ eHw(t) ≤ eHe−tA(αφ), t > 1.

In particular, Lemma 20.8 implies ‖uα(t)‖∞ ≤ Ct−η for all t > 0 provided η < n/2.The proof of H < ∞ above is based on [191, Lemma 2.3].

(iii) Let p = pS and φ be as in Theorem 28.10. Then uα∗ exists globally (see[232]) so that either its time decay is slower than the self-similar one or the solutiondoes not decay at all. Remark 22.10(ii) suggests that both possibilities can occur.Assume in addition that φ has only finitely many local minima and belongs tothe energy space u ∈ Lp+1(Rn) : |∇u| ∈ L2(Rn). Then [423] guarantees thatlimt→∞ ‖uα∗(t)‖∞tβ = ∞ and all non-threshold global solutions satisfy (28.21).

(iv) Let p > pS and let φ ∈ L∞ ∩ H1(Rn) satisfy the assumptions in Theo-rem 28.10. If Tmax(αφ) = ∞, then [356] implies ‖uα(t)‖∞ ≤ Ct−n/4 for t > 1.Since n/4 > β, the threshold solution uα∗ has to blow up in finite time.

In the proof of Theorem 28.10 we will need the following result on stationarysolutions of the rescaled equation (see [269], [412], [538], [163] and [390]).

Proposition 28.12. Let p > 1, λ ≥ 0 and let wλ = wλ(ρ) be the solution of theproblem

w′′ +n− 1

ρw′ +

ρ

2w′ + βw + |w|p−1w = 0 for ρ > 0, w(0) = λ, w′(0) = 0.

Then wλ is defined for all ρ > 0 and there exists finite limρ→∞ wλ(ρ)ρ2β =: A(λ).Given λ > 0, set ρλ := supρ > 0 : wλ > 0 on [0, ρ). Then the following is true:(i) If p ≤ pF and λ > 0, then ρλ < ∞.(ii) If pF λ0. In addition, A(λ) > 0 for λ ∈ (0, λ0) and A(λ0) = 0.(iii) If p ≥ pS, then ρλ = ∞ and A(λ) > 0 for all λ > 0.(iv) If p ≥ pJL, then the mapping λ → wλ(ρ) is strictly increasing for each fixedρ > 0 and supλ A(λ) = cp, where cp is the constant from (3.9).

Proof of Theorem 28.10. We have α∗ > 0 due to Theorem 20.6 and α∗ < ∞due to Theorem 17.1.

Since the solutions uα are radial we will consider them as functions uα(t) =uα(r, t), where r = |x|. Set v(ρ, s) = eβsu(es/2ρ, es − 1), ρ, s ≥ 0. Then v solvesthe equation

vs − vρρ −n− 1

ρvρ =

ρ

2vρ + βv + vp, (28.25)

cf. (18.13), (18.14).(i) Assume p < pS . Theorem 26.9 guarantees that any global positive radial

solution u = u(r, t) satisfies

‖u(t)‖∞ ≤ C0t−β , where C0 = C0(n, p).

This estimate, continuous dependence on initial data and the definition of α∗ showthat the solution uα∗ is global and satisfies the upper bound in (28.20).

If u is a solution (18.1), then the rescaled solution v of (28.25) satisfies

‖v(s)‖∞ = (t + 1)β‖u(t)‖∞, t = es − 1, (28.26)

hence‖v(s)‖∞ ≤ C0

(1 +

1t

)β

for all s ≥ 0, (28.27)

whenever u is global positive and radial. Since α∗φ ∈ L∞(Rn), the solution uα∗

remains bounded in L∞(Rn) on a small time interval. Now using (28.26) and(28.27) we can find C1 > 0 such that

‖vα∗(s)‖∞ < C1 for all s ≥ 0. (28.28)

Let λ0 be from Proposition 28.12 and fix λ ∈ (0, λ0). Then A := A(λ) > 0. Fixa ∈ (0, A) and set Wa(ρ) := aρ−2β . Choose δ > 0 such that

Wa(δ) > C1 + 1. (28.29)

An easy computation shows that the function V (ρ) = Va(ρ) := Wa(ρ − R1) is asupersolution of (28.25) for ρ ≥ R1 + δ provided R1 > 0 is large enough. In fact,

Vρρ +n− 1

ρVρ +

ρ

2Vρ + βV + V p ≤ Vρρ +

ρ

2Vρ + βV + V p

= a(ρ−R1)−2β−2[2β(2β + 1) + ap−1 − βR1(ρ−R1)

]< 0,

provided ρ ≥ R1 + δ and R1 > (2β(2β + 1) + ap−1)/βδ. Increasing R1 if necessarywe may also assume

V (ρ) > (α∗ + 1)φ(ρ) for all ρ ≥ R1, (28.30)

due to (28.19). Fix R2 > R1 + δ such that

wλ(ρ) > V (ρ) for ρ ≥ R2

where wλ is the solution from Proposition 28.12. We will show that vα∗(·, s) andwλ intersect in [0, R2] for any s ≥ 0, cf. Figure 14. This intersection guaranteesthe lower estimate in (28.20).

0 R1 + δ R2

C1 + 1

λwλ

Vvα∗(·, s)

Figure 14: Intersection of vα∗(·, s) and wλ in [0, R2].

Assume on the contrary that vα∗(ρ, s0) < wλ(ρ) for some s0 ≥ 0 and all ρ ∈[0, R2] and set

ε := inf[0,R2]

(wλ − vα∗(·, s0)

)> 0.

We have ‖vα(s) − vα∗(s)‖∞ < min(ε, 1) for all s ≤ s0 and α close to α∗, dueto the continuous dependence of solutions u of (18.1) on initial data. Fix suchα ∈ (α∗, α∗ + 1). Then

vα(ρ, s0) < wλ(ρ), ρ ∈ [0, R2], (28.31)

andvα(ρ, s) < C1 + 1, ρ ∈ [0, R2], s ≤ s0. (28.32)

Since vα(ρ, 0) = αφ(ρ) < V (ρ) for ρ ≥ R1 due to (28.30) and vα(R1 + δ, s) <C1 +1 < V (R1 +δ) for s ≤ s0 due to (28.32) and (28.29), the comparison principle(see Proposition 52.6) implies

vα(ρ, s) ≤ V (ρ) for ρ ≥ R1 + δ, s ≤ s0. (28.33)

Since V (ρ) < wλ(ρ) for ρ ≥ R2, estimates (28.33) and (28.31) imply vα(s0) < wλ,hence vα exists globally due to the comparison principle. But this contradicts thechoice of α∗ and concludes the proof of (28.20).

Next choose α ∈ (0, α∗]. Since vα is uniformly bounded due to vα ≤ v∗α and(28.28), the ω-limit set of vα(0, s)s≥0 is a compact interval J ⊂ [0, C1]. Assumethat J is not a singleton and fix λ ∈ (inf J, sup J) \ λ0. Then there exist aninfinite sequence s1 < s2 < s3 < . . . such that vα(0, sk) = λ for k = 1, 2, . . . .If λ > λ0, then wλ(ρλ) = 0 and the zero number z[0,ρλ](vα(s) − wλ) is finite fors > 0. However, this number has to drop at each sk, which yields a contradiction.Consequently, λ ∈ (0, λ0). Let A := limρ→∞ wλ(ρ)ρ2β , a ∈ (0, A), and let V = Va,δ and R1 be as above. Then vα(ρ, s) < V (ρ) for ρ ≥ R1 + δ and any s. FixR2 > R1 + δ such that wλ(R2) > V (R2). Then we obtain the same contradictionas above by considering the zero number z[0,R2](vα(s) − wλ). Consequently, thereexists λ = λ(α) ≥ 0 such that

vα(0, s)→ λ as s →∞. (28.34)

Due to the parabolic estimates the trajectory vα(s)s≥0 is relatively compact inC(R+) (considered with the locally uniform convergence) and its ω-limit set ωα

is a nonempty compact connected set, invariant under the semiflow generated by(28.25). In addition, (28.34) implies ψ(0) = λ for any ψ ∈ ωα. Assume that thereexists ψ ∈ ωα \ wλ and consider the solution v = vψ of (28.25) with initialdata ψ. Fix ρ0 > 0 and s0 > 0 such that vψ(ρ0, s) = wλ(ρ0) for all s ∈ [0, s0].Then the zero number z[0,ρ0](vψ(s) − wλ) is finite for s > 0 and has to dropat each s ∈ (0, s0) (due to vψ(0, s) = λ = wλ(0)) which yields a contradiction.

Consequently, ωα = wλ. Since vα ≥ 0 we have λ ≤ λ0. Similarly, estimates ofthe form vα(s) ≤ V for ρ ≥ R1 + δ show λ /∈ (0, λ0). Hence, λ(α) ∈ 0, λ0 for anyα ∈ (0, α∗].

Given 0 < α1 < α2 ≤ α∗, the function v = (α2/α1)vα1 is a subsolution of (28.25)and v(·, 0) = vα2(·, 0), hence vα2 ≥ v. Consequently, λ(α2) ≥ (α2/α1)λ(α1). Thisinequality guarantees λ(α) = 0 for all α < α∗ (and λ(α∗) = λ0). Hence, givenα < α∗, we have vα(s) → w0 = 0 locally uniformly in [0,∞) as s → ∞ andthe estimate vα(s) ≤ V on [R1 + δ,∞) concludes the proof of ‖vα(s)‖∞ → 0.Consequently, (28.21) is true.

(ii) Assume p ≥ pS. If α ∈ (0, α∗), then our assumptions imply the existence ofC1 > 0 such that the rescaled solution vα satisfies ‖vα(s)‖∞ < C1 for all s ≥ 0. Nowthe same arguments as in the proof of (i) show the existence of λ ∈ [0, C1] such that‖vα(s) − wλ‖∞ → 0 as s → ∞, where wλ is the solution from Proposition 28.12.However, for any a ∈ (0, 1) we have an estimate of the form

vα(ρ, s) ≤ a(ρ−R1)−2β , ρ > R1 + δ,

for some R1 = R1(a) > 0 (cf. (28.33)). In particular, assuming λ > 0, the choicea < A(λ) leads to a contradiction. Hence λ = 0 and (28.21) is true.

Finally consider the threshold solution uα∗ and assume on the contrary that‖vα∗(s)‖∞ ≤ C1 for all s ≥ 0. Then the arguments above guarantee

‖vα∗(s)‖∞ → 0 as s →∞. (28.35)

Fix λ > 0, a ∈ (0, A(λ)) and choose δ, R1 and R2 as in the proof of (i). Then thesame arguments as in that proof show that v∗α(s) and wλ intersect in [0, R2] forall s ≥ 0, which contradicts (28.35).

29. Decay and grow-up of threshold solutions inthe super-supercritical case

In this section we consider positive solutions of the Cauchy problem

ut −∆u = up, x ∈ Rn, t > 0,

u(x, 0) = u0(x), x ∈ Rn,

(29.1)

where n ≥ 11 and p > pJL. Set

m := 2/(p− 1)

and let U∗(r) = cpr−m be the singular stationary solution defined in (3.9). We will

use matched asymptotics to study the asymptotic behavior of solutions of (29.1)with initial data u0 ∈ L∞(Rn) satisfying

0 ≤ u0(x) ≤ U∗(|x|) for x = 0 (29.2)

and

U∗(|x|) − c1|x|− ≤ u0(x) ≤ U∗(|x|)− c2|x|− for |x| > c3, (29.3)

for some c1, c2, c3 > 0 and > m. Note that solutions u with such initial data areglobal (due to Theorem 20.5) and they are also threshold solutions in the sense ofSubsections 22.3, 28.4 since Tmax(λu0) <∞ for λ > 1 (due to [260]). Set

λ± :=12(n− 2− 2m±

√(n− 2− 2m)2 − 8(n− 2−m)

).

Due to Remark 9.4 and (9.4) there exists a > 0 such that the positive radial steadystate Uα = Uα(r) of (29.1) satisfying Uα(0) = α > 0 has the asymptotic expansion

U(r) = U∗(r) − aαr−m−λ− + o(r−m−λ−) as r →∞, (29.4)

where aα := α−λ−/ma. We will sketch the proof of the following theorem due to[203], [189], [204].

Theorem 29.1. Let p > pJL, ∈ (m, m + λ+ + 2). Suppose that u0 ∈ L∞(Rn)satisfies (29.2) and (29.3). Then there exist positive constants C1, C2 such that thesolution of (29.1) satisfies

C1(t + 1)α ≤ ‖u(·, t)‖∞ ≤ C2(t + 1)α for all t ≥ 0, (29.5)

where α := m(−m− λ−)/(2λ−).

Remarks 29.2. (i) The above theorem shows that threshold solutions can decayto zero with an arbitrarily slow decay rate (if ∈ (m, m + λ−)) and also can growup with any rate of the form tα, α ∈ (0, m(2+λ+−λ−)/(2λ−)). The upper boundfor α is known to be optimal. More precisely, if p > pJL and u0 ∈ L∞(Rn) satisfies(29.2) (but not necessarily (29.3)), then u is global and ‖u(t)‖∞ ≤ C(t + 1)α∗

,where α∗ = m(2+λ+−λ−)/(2λ−). In addition, there exists u0 ∈ L∞(Rn) satisfying(29.2) such that ‖u(t)‖∞ ≥ c(t + 1)α∗

(see [381]).(ii) Let p = pJL. Then λ− = λ+ =: λ. If u0 ∈ L∞(Rn) satisfies (29.2) and

(29.3) with some ∈ (m + λ, m + λ + 2), then (29.5) remains true with ‖u(·, t)‖∞replaced by ‖u(·, t)‖∞

(log(t + 2)

)−m/λ, see [190].(iii) Assume psg 0.

This is a consequence of [232, Theorem 10.1(i)] (see also [499]). Therefore thecondition p ≥ pJL, for grow-up or slow decay below the singular steady-state, isoptimal.

The idea of matched asymptotics is to find a suitable asymptotic expansion forthe solution in an inner region (for “small” |x|) and an outer region (for “large” |x|).

Matching these expansions on the boundary of the inner and outer regions (thatis, comparing the coefficients of the leading terms of the expansions) determinesthe quantity that we are looking for. This formal approach not only providesa guess for the behavior of solutions but often also suggests the form of sub-and supersolutions that enable one to prove the result rigorously. It should bementioned that in many cases the approach is more complicated: For example, inaddition to the inner and outer regions one also has to consider an intermediateregion.

We will only consider the case < m + λ− in Theorem 29.1 since the case >m+λ− can be treated by similar arguments and the proof in the case = m+λ−follows from the fact that the solution remains between two positive stationarysolutions of (29.1) for t ≥ t0 > 0 due to the comparison principle and (29.4). Inaddition, we will only describe in detail the formal part of the proof; the rigorouspart will be sketched. Although the detailed rigorous proof in [204] representsone of the simplest applications of matched asymptotics, it is still quite long andtechnical and lies beyond the scope of this book. Another relatively simple exampleof matched asymptotics is mentioned in Remark 40.9(c).

Throughout the rest of this section we will write f ∼ g if C1g ≤ f ≤ C2g forsome constants C1, C2 > 0 and f ≈ g (or f = g + h.o.t.) if f − g = o(f).

Sketch of proof of Theorem 29.1 for < m + λ−.Part 1: Formal matched asymptotics. We will consider radial solutions u =

u(r, t), r = |x| of (29.1). Such solutions satisfy

ut = urr +n− 1

rur + up, r > 0, t > 0,

u(r, 0) = u0(r), r > 0.

⎫⎬⎭ (29.6)

Assume that u0 is continuous and radial nonincreasing and that

η(t) := u(0, t) behaves like (t+1)α for some α ∈ (−m/2, 0) and t 1. (29.7)

Notice that introducing a new variable ζ = ζ(t, r) := η1/m(t)r and assuming thatu can be written in the form u = η(t)ϕ(ζ), (29.6) is transformed to

ηtη−p(ϕ +

1m

ζϕζ

)= ϕζζ +

n− 1ζ

ϕζ + ϕp,

where ηtη−p → 0 as t →∞. Consequently,

the solution u should asymptotically behave like η(t)ϕ(η(t)1/mr), (29.8)

where ϕ is a solution of

ϕζζ +n− 1

ζϕζ + ϕp = 0, ζ > 0, ϕ(0) = 1, ϕζ(0) = 0. (29.9)


It turns out that before making the transformation mentioned above it is usefulto apply the self-similar change of variables

v(ρ, s) = (t + 1)m/2u(r, t), ρ =r√

t + 1, s = log(t + 1),

which transforms (29.6) into

vs = vρρ +n− 1

ρvρ + vp +

ρ

2vρ +

m

2v, ρ > 0, s > 0,

v(ρ, 0) = v0(ρ) := u0(ρ), ρ > 0.

⎫⎬⎭ (29.10)

Notice that v(0, s) = (t + 1)m/2u(0, t)→∞ as s →∞ due to (29.7).Let us first consider the inner region (where ρ is small). The equation in (29.10)

is “similar” to that in (29.6) for small ρ: The additional two terms at the end ofthe RHS are expected to be small in comparison to the remaining ones if v is largeand ρ small. Therefore, taking into account (29.8) and (29.9), for small ρ we willlook for solution v in the form

v(ρ, s) = σ(s)(ψ(ξ) −R(s, ξ)

)(29.11)

where σ(s) := v(0, s), ξ := σ1/mρ, ψ is the solution of

ψξξ +n− 1

ξψξ + ψp = 0, ξ > 0, ψ(0) = 1, ψζ(0) = 0, (29.12)

and R represents the higher order terms (remainder). Plugging the ansatz (29.11)into (29.10) we obtain R ≈ σsσ

−pΨ(ξ) for ρ small and s large, where

Ψξξ +n− 1

ξΨξ + pψp−1Ψ =

(mσ

2σs− 1)(

ψ +1m

ξψξ

), ξ > 0,

Ψ(0) = Ψξ(0) = 0.

⎫⎬⎭ (29.13)

Since we expect σ(s) to behave like e(m/2+α)s for some α ∈ (−m/2, 0) due to(29.7), the coefficient ( mσ

2σs− 1) in (29.13) behaves like a positive constant and

[204, Lemma 3.1], [203, Lemma 4.2] guarantee that there exists K > 0 such thatΨ(ξ) ≈ Kξ2−m−λ− as ξ → ∞. Fixing ρ > 0, we have ξ = σ1/m(s)ρ → ∞ ass →∞, hence

R(s, ξ) ≈ σs

σpΨ(ξ) ≈ K1

1σp−1

Ψ(ξ) ≈ K2ξ−2Ψ(ξ) ≈ K3ξ

−m−λ− ,

where K1, K2, K3 are positive constants. Due to (29.4), the solution ψ of (29.12)satisfies

ψ(ξ) = cpξ−m − aξ−m−λ− + o(ξ−m−λ−), as ξ →∞,

where a > 0. Consequently, we obtain the two-term inner expansion

v ≈ σ(cpξ−m − aξ−m−λ−) = cpρ

−m − aσ−λ−/mρ−m−λ− , (29.14)

where a = a + K3 > 0.Next we consider the formal expansion in the outer region (where ρ 1) as

s →∞. Settingv = cpρ

−m − w

and assuming w ρ−m for ρ 1, we have

ws = wρρ +n− 1

ρwρ +

pcp−1p

ρ2w +

ρ

2wρ +

m

2w + h.o.t., ρ 1.

If we look for a solution w in the form

w(ρ, s) = e−βsW (ρ) + h.o.t.,

then W has to solve the equation

Wρρ +n− 1

ρWρ +

pcp−1p

ρ2W +

ρ

2Wρ +

(β +

m

2

)W = 0. (29.15)

In addition, due to our assumption (29.3), W is required to satisfy the condition

0 < lim infρ→∞ ρ W (ρ) ≤ lim sup

ρ→∞ρ W (ρ) < ∞. (29.16)

If ρ 1, then the last two terms in (29.15) are much greater than the remainingones so that we have to guarantee ρ

2Wρ ≈ −(β + m

2

)W . Due to (29.16) we have

to set β := ( −m)/2. In order that the outer expansion matches with the innerexpansion (29.14), W should also satisfy

0 < lim infρ→0

ρm+λ−W (ρ) ≤ lim supρ→0

ρm+λ−W (ρ) <∞. (29.17)

It is known (see [1] or [204]) that the problem (29.15), (29.16), (29.17) with β =( − m)/2 possesses a positive solution W provided ∈ (m, m + λ+ + 2) (thissolution can be expressed explicitly in terms of Kummer’s functions). Hence, weobtain the two-term outer expansion

v ≈ cpρ−m − e−( −m)s/2W (ρ). (29.18)

If we now match the inner expansion (29.14) with the outer expansion (29.18)at ρ = ρ0 > 0, then we obtain

σ(s) ∼ em( −m)s/(2λ−), (29.19)

hence

u(0, t) ∼ tα, where α =m(−m− λ−)

2λ−.

This gives a formal proof of Theorem 29.1 for < m + λ−.

Part 2: Sketch of the rigorous proof. We will find a subsolution v and a super-solution v for the solution v of (29.10) such that the estimates v ≤ v ≤ v willguarantee (29.5).

It is relatively easy to check that the subsolution v can be chosen as

v(ρ, s) := max(0, cpρ

−m − be−( −m)s/2W (ρ)),

where W is a fixed solution of (29.15), (29.16), (29.17) with β = ( −m)/2, andb > 0 is large enough.

The supersolution v is defined by

v(ρ, s) :=

v1(ρ, s), s ≥ 0, ρ ≤ ρM (s),v2(ρ, s), s ≥ 0, ρ > ρM (s),

where ρM (s) := infρ > 0 : v2(ρ, s) < v1(ρ, s) and v1, v2 are supersolutions inthe corresponding domains.

It is again relatively easy to check that the supersolution v2 can be chosen inthe form

v2(ρ, s) := cpρ−m − be−( −m)s/2W (ρ),

where W is the solution of

Wρρ +n− 1

ρWρ +

ρ

2Wρ +

2W = 0, ρ > 0, W (0) = 1, Wρ(0) = 0,

(which can be again expressed in terms of Kummer’s functions) and b is smallenough.

The most difficult part is the choice of the supersolution v1. Recall that in theinner region, we expect

v(ρ, s) ≈ σ(s)(ψ(ξ)− σs

σpΨ(ξ)

),

where Ψ solves (29.13). Plugging (29.19) into (29.13) we see that Ψ solves theproblem

Ψξξ +n− 1

ξΨξ + pψp−1Ψ =

m + λ− −

−m

(ψ +

1m

ξψξ

)+ RΨ, ξ > 0,

Ψ(0) = Ψξ(0) = 0,

⎫⎬⎭ (29.20)

where RΨ represents higher order terms. Now it turns out that one can set

v1(ρ, s) := σ(s)(ψ(ξ)− σs

σpΨ(ξ)

),

where Ψ is the solution of (29.20) with RΨ := A/(1 + ξm+λ−) and A is a suitablepositive constant. (The term RΨ is purely technical.)

Chapter III

Systems

30. Introduction

Chapter III is devoted to systems of elliptic and parabolic types. In Section 31,we study the questions of a priori estimates and existence for weakly coupledelliptic systems which are natural extensions of the scalar equations considered inChapter I. In Section 32, we study a simple parabolic system which is the analogueof the scalar model problem (15.1) studied in Chapter II. For this system, we treatthe questions of well-posedness, global existence and blow-up. In Section 33, wediscuss the different possible effects of adding linear diffusion (and some boundaryconditions) to a system of ODE’s. It will turn out that quite opposite effects canbe observed. This will lead us to consider some systems arising in biological orphysical contexts, such as mass-preserving and Gierer-Meinhardt systems.

31. Elliptic systems

In Sections 10–13, we have studied several methods to derive a priori estimates ofpositive solutions of scalar elliptic equations. The aim of this section is to presentanalogous results and methods in the case of elliptic systems. The three methodsthat we shall describe are extensions of the methods of Sections 11–13 from thescalar case, but they require substantial additional work and several new ideas. Asmentioned before, a priori estimates can be used for the proof of existence, andthey do not require any variational structure of the problem. Therefore they arewell-suited for elliptic systems, which do not possess such structure in general.

We will devote our attention to the Dirichlet problem for superlinear systems,especially of cooperative type, of the form:

−∆u = f(x, u, v), x ∈ Ω,

−∆v = g(x, u, v), x ∈ Ω,

u = v = 0, x ∈ ∂Ω.

⎫⎪⎬⎪⎭ (31.1)

A simple model case of such systems, and the analogue of the scalar problem(3.10), is the Lane-Emden system:

−∆u = vp, x ∈ Ω,

−∆v = uq, x ∈ Ω,

u = v = 0, x ∈ ∂Ω.

⎫⎪⎬⎪⎭ (31.2)

252 III. Systems

Throughout this section we assume p, q > 1, and we denote

α =2(p + 1)pq − 1

, β =2(q + 1)pq − 1

. (31.3)

These numbers play a fundamental role in the analysis of (31.2). They representscaling exponents, corresponding to the fact that, for each λ > 0, the differentialequations in (31.2) are invariant under the transformation (u, v) → (uλ, vλ), whereuλ(x) = λαu(λx), vλ(x) = λβv(λx), due to α + 2 = βp, β + 2 = αq. On the otherhand we say that (u, v) is positive if u, v > 0 (a.e.) in Ω. Note that, of course,if (u, v) is a nontrivial nonnegative, say classical, solution of (31.2) in a domainΩ ⊂ R

n, then it is positive by the strong maximum principle.

Remarks 31.1. (i) Other nonlinearities. Although we shall concentrate, forsimplicity, on the model case (31.2) and on a few variants, the three methodsthat we describe below, or their modifications, can be applied to wide varieties ofsystems. Let us mention systems with products or sums of powers, respectivelygiven by

f = urvp, g = vsuq (31.4)(see [371], [454], [136], [449]), and by

f = ur + vp, g = vs + uq

(see [184], [547], [449]), with p, q, r, s > 0. Several systems arising in physical orbiological applications are also tractable by these methods. Let us mention thecooperative logistic system given by

f = auv + u(c− u), g = buv + v(d− v)with a, b, c, d > 0 constants (see e.g. [343] and the references therein), which arisesin population dynamics, where u, v stand for the densities of two biological species.Another example is given by

f = uv − au, g = bu (31.5)with a, b > 0 constants (see [258], [122], [449]), which arises as a model of nuclearreactor, where u and v respectively represent the neutron flux and the temperature.Each of the three methods works under different (and generally non-comparable)sets of assumptions, and its applicability depends on the problem under consider-ation (see Theorem 31.17 for an example in the case of (31.5)).

(ii) Noncooperative systems. Many interesting examples from the point ofview of biological or chemical applications involve noncooperative systems or sys-tems with balance law. Results and techniques concerning the questions of globalexistence and blow-up for the parabolic version of such systems are presented inSection 33 below.

(iii) Singularities for elliptic systems. Some results on isolated singularitiesfor systems (31.2) and (31.1), (31.4), extending those in Section 4, can be foundin [82], [80], [424].

31. Elliptic systems 253

31.1. A priori bounds by the method of moving planes andPohozaev-type identities

We consider the Lane-Emden system (31.2). For this system, the method describedin this subsection allows to obtain complete and optimal results in the case ofconvex domains.

Theorem 31.2. Assume p, q > 1, Ω convex and bounded, and

1p + 1

+1

q + 1>

n− 2n

, (31.6)

equivalentlyα + β > n− 2.

(i) Then any positive classical solution of (31.2) satisfies the a priori estimate

‖u‖∞, ‖v‖∞ ≤ C, (31.7)

with C independent of (u, v).(ii) There exists a positive classical solution of (31.2).

Theorem 31.3. Assume p, q > 1, n ≥ 3, Ω starshaped and bounded, and

1p + 1

+1

q + 1≤ n− 2

n, (31.8)

equivalentlyα + β ≤ n− 2.

Then (31.2) has no positive classical solution.

Theorems 31.2 and 31.3 are respectively due to [134] (see also [413]) and to[370]. The critical curve in the (p, q) plane:

1p + 1

+1

q + 1=

n− 2n

,

associated with condition (31.6), is called the Sobolev hyperbola. Note that inthe scalar case, corresponding to p = q, condition (31.6) reduces to p < pS .

The method of proof of Theorem 31.3 is a modification of that of Section 13 inthe scalar case. A common ingredient to the proofs of Theorems 31.2 and 31.3 isthe following variational identity of Pohozaev-type [370], which is the analogue ofTheorem 5.1 in the scalar case.

254 III. Systems

Lemma 31.4. Assume Ω bounded.(i) For any functions u, v ∈ C2(Ω) such that u = v = 0 on ∂Ω, there holds∫

Ω

[(x · ∇v)∆u + (x · ∇u)∆v − (n− 2)∇u · ∇v

]dx =

∫∂Ω

(x · ν)∂u

∂ν

∂v

∂νdσ.

(ii) For any nonnegative classical solution (u, v) of (31.2) and any θ ∈ [0, 1], thereholds ∫

Ω

[( n

p + 1− (n− 2)θ

)vp+1 +

( n

q + 1− (n− 2)(1− θ)

)uq+1

]dx

=∫

∂Ω

(x · ν)∂u

∂ν

∂v

∂νdσ.

(31.9)

Proof. (i) We compute

div((x · ∇v)∇u) = (x · ∇v)∆u + (∇(x · ∇v) · ∇u)

= (x · ∇v)∆u +∑i,j

∂

∂xi

(xj

∂v

∂xj

) ∂u

∂xi

= (x · ∇v)∆u +∑i,j

xj∂2v

∂xi∂xj

∂u

∂xi+∑

i

∂v

∂xi

∂u

∂xi.

Therefore

div[(x · ∇v)∇u + (x · ∇u)∇v

]= (x · ∇v)∆u + (x · ∇u)∆v + x · ∇(∇u · ∇v) + 2∇u · ∇v.

On the other hand, we have

div[x(∇u · ∇v)

]= (div x) (∇u · ∇v) + x · ∇(∇u · ∇v)

= n(∇u · ∇v) + x · ∇(∇u · ∇v).

By subtracting, we obtain

div[(x · ∇v)∇u + (x · ∇u)∇v − x(∇u · ∇v)

]= (x · ∇v)∆u + (x · ∇u)∆v − (n− 2)∇u · ∇v.

Applying the divergence theorem, it follows that∫Ω

[(x · ∇v)∆u + (x · ∇u)∆v − (n− 2)∇u · ∇v

]dx


=∫

∂Ω

[(x · ∇v)∇u + (x · ∇u)∇v − x(∇u · ∇v)

]· ν dσ.

Since ∇u = (∂u∂ν ) ν, ∇v = ( ∂v

∂ν ) ν on ∂Ω, due to u = v = 0 on ∂Ω, assertion (i)follows.

(ii) For a solution (u, v) of (31.2), we have

(x · ∇v)∆u + (x · ∇u)∆v = −(x · ∇v) vp − (x · ∇u)uq

= −x · ∇( vp+1

p + 1+

uq+1

q + 1

)= −div

(x( vp+1

p + 1+

uq+1

q + 1

))+ n( vp+1

p + 1+

uq+1

q + 1

)hence ∫

Ω

[(x · ∇v)∆u + (x · ∇u)∆v

]dx = n

∫Ω

( vp+1

p + 1+

uq+1

q + 1

)dx. (31.10)

On the other hand,∫Ω

∇u · ∇v dx = −∫

Ω

u ∆v dx =∫

Ω

uq+1 dx

and ∫Ω

∇u · ∇v dx = −∫

Ω

v ∆u dx =∫

Ω

vp+1 dx

yield ∫Ω

∇u · ∇v dx =∫

Ω

((1− θ)uq+1 + θ vp+1

)dx. (31.11)

In view of (i), assertion (ii) then follows by combining (31.10) and (31.11).

We first prove Theorem 31.3, which follows easily from Lemma 31.4.

Proof of Theorem 31.3. In view of (31.8), by choosing θ = n(n−2)(p+1) ∈ (0, 1),

we getn

p + 1− (n− 2)θ = 0,

n

q + 1− (n− 2)(1− θ) ≤ 0. (31.12)

Identity (31.9) in Lemma 31.4 then implies∫

∂Ω(x · ν) ∂u∂ν

∂v∂ν dσ ≤ 0. Now since Ω

is starshaped around, say, x = 0, we have x · ν ≥ 0 on ∂Ω, along with ∂u∂ν , ∂v

∂ν ≤ 0,hence ∫

∂Ω

(x · ν)∂u

∂ν

∂v

∂νdσ = 0. (31.13)

If the inequality in (31.8) is strict, then so is the inequality in (31.12) and wededuce from (31.9) that u ≡ 0, hence v ≡ 0. In the equality case, then since

256 III. Systems

x · ν ≡ 0 on ∂Ω, (31.13) implies ∂u∂ν = 0 or ∂v

∂ν = 0 at some point of ∂Ω. Since−∆u,−∆v ≥ 0, u, v ≥ 0 in Ω and u = v = 0 on ∂Ω, we infer from Hopf’s lemmathat u ≡ 0 or v ≡ 0, hence u ≡ v ≡ 0. (Note that this last argument actuallyapplies whenever (31.8) holds.)

Proof of Theorem 31.2. (i) It is more involved and requires several steps.Step 1. Basic L1

loc estimates. We claim that∫Ω

uϕ1 dx ≤ C,

∫Ω

vϕ1 dx ≤ C. (31.14)

Multiplying by ϕ1, integrating by parts, and using Jensen’s inequality, we obtain

λ1

∫Ω

uϕ1 dx =∫

Ω

vpϕ1 dx ≥(∫

Ω

vϕ1 dx)p

andλ1

∫Ω

vϕ1 dx =∫

Ω

uqϕ1 dx ≥(∫

Ω

uϕ1 dx)q

.

Consequently, we have (∫Ω

uϕ1 dx)pq

≤ λp+11

∫Ω

uϕ1 dx,

which yields the first inequality in (31.14). The second follows similarly.Step 2. Estimates near ∂Ω. We use the notation of Section 13 (see after Theo-

rem 13.1). Since Ω is convex and smooth, we can find λ0, c0 > 0 such that

Σ′(y, λ) ⊂ Ω, λ ≤ λ0 and (ν(x), ν(y)) > c0, x ∈ ∂Σ(y, λ0) ∩ ∂Ω.

Similarly as in Theorem 13.1, we shall apply the moving planes method (cf. [514]in the case of systems) to show that

u(R(y, λ)x) ≥ u(x), v(R(y, λ)x) ≥ v(x), y ∈ ∂Ω, x ∈ Σ(y, λ), λ ≤ λ0.(31.15)

Without loss of generality, we may assume that y = 0 and that ν(0) = −e1. Foreach x = (x1, x

′), we denote xλ := R(0, λ)x = (2λ − x1, x′), Σλ := Σ(0, λ) =

Ω ∩ x1 < λ, and Σ′λ := Σ′(0, λ) = R(0, λ)Σλ. Define

wλ(x) = u(xλ)− u(x), zλ(x) = v(xλ)− v(x), for x ∈ Σλ, 0 < λ ≤ λ0,

and set

E :=µ ∈ (0, λ0] : wλ(x) ≥ 0, zλ(x) ≥ 0 for all x ∈ Σλ and λ ∈ (0, µ)

.

Since ∂u∂x1

(0) > 0, ∂v∂x1

(0) > 0 by Hopf’s lemma, we have λ ∈ E for λ > 0 small.Assume for contradiction that λ := sup E < λ0. We have

wλ ≥ 0, zλ ≥ 0, for all x ∈ Σλ and λ ∈ (0, λ], (31.16)

and there exists a sequence λi → λ, with λ < λi < λ0, such that (for instance)minΣλi

wλi < 0. Since wλ = 0 on x1 = λ ∩ Ω and

wλ > 0 on x1 < λ ∩ ∂Ω, for all λ < λ0, (31.17)

it follows that this minimum is attained at a point qi ∈ Σλi . Therefore ∇wλi (qi) =0. On the other hand, since ∂u

∂x1= (e1 · ν) ∂u

∂ν ≥ c > 0 on x1 ≤ λ0 ∩ ∂Ω and

wλ(x) = u(2λ− x1, x′)− u(x1, x

′) = 2(λ− x1)∂u

∂x1(ξ(x)),

with |ξ(x) − x| ≤ 2(λ− x1), we see that wλ(x) ≥ 0 for x in an ε-neighborhood ofx1 = λ ∩ ∂Ω, with ε > 0 independent of λ ∈ (0, λ0]. Therefore, we may assumethat qi → q ∈ Σλ, q /∈ x1 = λ ∩ ∂Ω, and by continuity we get

wλ(q) = 0 and ∇wλ(q) = 0. (31.18)

But (31.16) implies

−∆wλ(x) = vp(xλ)− vp(x) ≥ 0 and wλ(x) ≥ 0, x ∈ Σλ.

By Hopf’s lemma, this along with (31.18) implies wλ = 0 in Σλ, contradicting(31.17). Consequently, λ = λ0, which proves (31.15). This guarantees that

(∇u(x), ν(y)) ≤ 0, (∇v(x), ν(y)) ≤ 0, y ∈ ∂Ω, x ∈ Σ(y, λ0). (31.19)

By Lemma 13.2 and Step 1, we deduce that u, v ≤ C on Ωε = z ∈ Ω : δ(z) < ε forsome ε, C > 0 depending only on Ω. Using interior-boundary elliptic Lp-estimates(see Appendix A) and the embedding W 2,k → BUC1 for k > n, we deduce auniform bound for ∇u,∇v in Ωε/2. In particular, we have shown that

∣∣∣∂u

∂ν

∣∣∣, ∣∣∣∂v

∂ν

∣∣∣ ≤ C, x ∈ ∂Ω. (31.20)

Step 3. Energy estimates. We claim that∫Ω

vp+1 dx ≤ C,

∫Ω

uq+1 dx ≤ C.

258 III. Systems

Since 1p+1 + 1

q+1 > n−2n , we may choose θ ∈ (0, 1), such that

n

p + 1− (n− 2)θ > 0,

n

q + 1− (n− 2)(1− θ) > 0.

Using assertion (i) of Lemma 31.4 and estimate (31.20), we deduce that∫Ω

vp+1 dx +∫

Ω

uq+1 dx ≤ C

∫∂Ω

(x · ν)∂u

∂ν

∂v

∂νdσ ≤ C.

Step 4. Bootstrap. Pick ρ > 1 to be fixed later and consider the following induc-tion hypothesis:

‖u‖(q+1)ρi , ‖v‖(p+1)ρi ≤ C. (Hi)

Step 3 guarantees that (H0) is verified. Assume that (Hi) holds for some i ∈ N.Then, since (u, v) solves (31.2), the linear estimate in Proposition 47.5(i) implies(Hi+1) provided

p

(p + 1)ρi− 1

(q + 1)ρi+1<

2n

andq

(q + 1)ρi− 1

(p + 1)ρi+1<

2n

.

It is thus sufficient that

p

(p + 1)− 1

(q + 1)ρ<

2n

andq

(q + 1)− 1

(p + 1)ρ<

2n

,

i.e.1ρ

> max[(q + 1)

(n− 2n

− 1p + 1

), (p + 1)

(n− 2n

− 1q + 1

)].

Since, by assumption, 1p+1 + 1

q+1 = n−2n + ε for some ε > 0, it suffices to choose

1ρ

> 1− ε min(p + 1, q + 1).

After a finite number of steps, we obtain ‖u‖q ≤ C, ‖v‖p ≤ C for some q > nq/2,p > np/2, and a further application of Proposition 47.5(i) yields ‖u‖∞ ≤ C,‖v‖∞ ≤ C.

(ii) The proof is similar to that of Corollary 10.3 (see e.g. [180] or [449, Section 4]for details).

Remarks 31.5. Limitations and extensions. (i) The above method does notextend to general systems of the form (31.1). Indeed (but for very special cases), fshould not depend on u (nor g on v) because of the need of variational identities.Also, f, g cannot depend on x (at least in an arbitrary way) in order to applythe moving planes method. It can still be generalized to f = f(v), g = g(u),with f, g nondecreasing (in order for the system to admit a comparison principle


to apply the moving planes method), provided f, g also satisfy suitable growthconditions related with the Sobolev hyperbola. These conditions can be expressedas a relation between f, g and their primitives which enables one to control

∫Ω

vf(v)and

∫Ω

ug(u) from the variational identities.(ii) The method partially extends to nonconvex domains Ω (via the Kelvin

transform). However, this requires additional growth restrictions if n ≥ 3, namelyp, q ≤ pS in the case of (31.2).

Remark 31.6. Variational methods. If the nonlinearities f, g in system (31.1)have the form f(x, u, v) = Hv(x, u, v), g(x, u, v) = Hu(x, u, v), then solutions of(31.1) can be found as critical points of the functional

Φ(u, v) :=∫

Ω

∇u · ∇v dx−∫

Ω

H(x, u, v) dx.

Considering Φ as a strongly indefinite functional in W 1,20 × W 1,2

0 (Ω) (or, moregenerally, in spaces of the form Xα × X1−α, where Xα, α ∈ (0, 1), are suitableinterpolation spaces between X0 := L2(Ω) and X1 := W 2,2 ∩W 1,2

0 (Ω), see [285]or [182], for example, and cf. Section 51.1) often leads to unnecessary technicalrestrictions concerning the growth of the Hamiltonian H . To overcome these diffi-culties one can use a dual approach (see [137] in the case of systems or [24], [25] inthe scalar case). In the particular case of the Lane-Emden system (31.2) we haveH = |v|p+1/(p + 1) + |u|q+1/(q + 1) and the dual functional has the form

Φ(w, z) =∫

Ω

( |w|p1

p1+|z|q1

q1− 1

2(K ∗ w)z

)dx,

where p1 = 1 + 1/p, q1 = 1 + 1/q, w = |v|p−1v, z = |u|q−1u and K is the Greenfunction for the negative Dirichlet Laplacian, that is u := K ∗w is the solution ofthe problem

−∆u = w in Ω, u = 0 on ∂Ω.

(Notice that∫Ω (K ∗ w)z dx =

∫Ω (K ∗ z)w dx.) The functional

Φ : Lp1 × Lq1(Ω) → R

possesses a mountain-pass structure and, in particular, it is easy to show that theexistence result in Theorem 31.2 remains true without the assumption Ω convex.However, this approach does not provide a priori estimates of solutions.

For some particular nonlinearities f, g, system (31.1) can also be reduced toa single higher-order equation. This is for instance the case for the Lane-Emdensystem (31.2), which is equivalent to the problem

−∆((−∆u)1/p) = uq, x ∈ Ω,

u = ∆u = 0, x ∈ ∂Ω,

where u ≥ 0 ≥ ∆u. Again, this problem can solved by variational methods.

260 III. Systems

31.2. Liouville-type results for the Lane-Emden system

In this subsection we state Liouville-type theorems for the Lane-Emden system(and prove some of them). These are statements about nonexistence of entirepositive solutions in the whole space or in a half-space. As in the scalar case, theyconstitute essential pieces of information in view of the rescaling method (see nextsubsection).

We thus consider the following problems:

−∆u = vp, x ∈ Rn,

−∆v = uq, x ∈ Rn,

(31.21)

or−∆u = vp, x ∈ R

n+,

−∆v = uq, x ∈ Rn+,

u = v = 0, x ∈ ∂Rn+,

⎫⎪⎬⎪⎭ (31.22)

where p, q > 1 and Rn+ := x ∈ R

n : xn > 0.

Conjecture 31.7. Systems (31.21) and (31.22) do not admit any positive classicalsolutions if (p, q) lies below the Sobolev hyperbola, i.e.

α + β > n− 2.

Remarks 31.8. “Classical solutions” in Conjecture 31.7 means u, v ∈ C2(Rn)and u, v ∈ C2(Rn

+)∩C(Rn+), respectively; no growth or decay conditions at infinity

are imposed. However for the rescaling method, it is sufficient to know a Liouville-type theorem for bounded positive solutions.

Although the full Conjecture 31.7 has not been proved so far, it is stronglysupported by the following results.

Theorem 31.9. Let p, q > 1.(i) Assume α + β ≤ n − 2. Then system (31.21) admits some radial, bounded,positive classical solution.(ii) System (31.21) does not admit any positive classical solution in the followingcases:

(a) α + β > n− 2 and either u, v are radial or n = 3,(b) max(α, β) ≥ n− 2,(c) p, q ≤ pS, (p, q) = (pS , pS).

Assertion (i) is due to [472]. As for assertion (ii), part (a) is due to [371] in theradial case. In the nonradial case, part (a) settles Conjecture 31.7 for n = 3. This


is due to [471] for polynomially bounded solutions, and to [424] in the general case.Part (b) is actually valid for supersolutions (see Theorem 31.12 below). Part (c),which in particular recovers the (optimal) scalar case, is due to [181]. The Liouville-type result is also known in some other parts of the region α+β > n−2 (see [106]).Since most of the proofs are long and technical, we shall only prove nonexistenceunder assumption (b), as a consequence of Theorem 31.12 below.

As for the half-space case, we have the following reduction. The result is due to[87], generalizing the idea of Theorem 8.3 in the scalar case.

Theorem 31.10. Let n ≥ 1. For given p, q > 1, if system (31.21) does not admitany bounded, positive classical solution in dimension n − 1, then system (31.22)does not admit any bounded, positive classical solution in dimension n.

Remarks 31.11. (i) Here we have made the convention that functions of n = 0variables are constants (and have null Laplacian). Consequently the conclusion ofTheorem 31.10 is true for n = 1.

(ii) If system (31.21) does not admit any bounded positive solution in dimensionn, then this remains true in dimension n− 1, so that in particular, system (31.22)does not admit any bounded positive solution in dimension n. Indeed if (u, v) solves(31.21) in R

n−1 and if we let u(x) = u(x1, . . . , xn−1), v(x) = v(x1, . . . , xn−1), then(u, v) solves (31.21) in R

n.(iii) On the other hand, it was shown in [424] that, for given p, q > 1 and n,

if system (31.21) does not admit any bounded, positive classical solution, then itdoes not admit any positive classical solution at all, and system (31.22) does notadmit any bounded, positive classical solution.

Sketch of proof of Theorem 31.10. Assume that (31.22) admits a boundedpositive solution (u, v). By using the moving planes method (see the proof ofTheorem 21.10 for similar arguments in the scalar case), one can show that ∂u

∂xn≥ 0

and ∂v∂xn

≥ 0 in Rn+. Therefore, for each x′ ∈ R

n−1,

U(x′) := limxn→∞u(x′, xn) and V (x′) := lim

xn→∞ v(x′, xn)

are well defined and are bounded positive functions. Arguing exactly as in theproof of Theorem 8.3, we see that (U, V ) is a bounded, positive classical solutionof system (31.21) in R

n−1. The result follows.

Case (b) of Theorem 31.9(ii) is actually true for the following system of inequal-ities (see [501], [372]):

−∆u ≥ vp, x ∈ Rn,

−∆v ≥ uq, x ∈ Rn.

(31.23)

262 III. Systems

Theorem 31.12. Let p, q > 1. System (31.23) does not admit any positive solu-tion u, v ∈ C2(Rn) if max(α, β) ≥ n− 2.

Proof. It is based on the rescaled test-function method. Fix φ ∈ D(Rn), 0 ≤ φ ≤1, such that φ(x) = 1 for |x| ≤ 1 and φ(x) = 0 for |x| ≥ 2. For each R > 0, putφR(x) = φ(x/R). Let m, k ≥ 2 to be fixed later. We note that ∆(φm

R ) = 0 for|x| ≤ R and that

|∆(φmR )| =

∣∣mφm−1R ∆φR + m(m− 1)φm−2

R

∣∣∇φR|2∣∣ ≤ CR−2φm−2

R .

Multiplying the first inequality in (31.23) by φmR and integrating by parts, we

obtain ∫vp φm

R ≤ −∫

φmR ∆u = −

∫u ∆(φm

R ) ≤ CR−2

∫R<|x|<2R

u φm−2R

(where∫

=∫

Rn). Applying Holder’s inequality, it follows that∫vp φm

R ≤ CR(n/q′)−2(∫

R<|x|<2R

uq φ(m−2)qR

)1/q

.

Similarly, we obtain∫uq φk

R ≤ CR(n/p′)−2(∫

R<|x|<2R

vp φ(k−2)pR

)1/p

.

Now, since p, q > 1, we have 2 + (k/q) < (k − 2)p for k large enough, and we canthen choose m such that 2 + (k/q) ≤ m ≤ (k − 2)p, that is: (k − 2)p ≥ m and(m− 2)q ≥ k. Therefore,

(∫vp φm

R

)pq

≤ CR((n/q′)−2)pq(∫

R<|x|<2R

uq φkR

)p

,

(∫uq φk

R

)p

≤ CR((n/p′)−2)p

∫R<|x|<2R

vp φmR .

Consequently, (∫vp φm

R

)pq

≤ CRθ

∫R<|x|<2R

vp φmR , (31.24)

where

θ = pq( n

q′−2)+p( n

p′−2)

= p(n(q−1)−2q)+n(p−1)−2p = (n−2)(pq−1)−2(p+1).

In particular, (∫|x|<R

vp)pq−1

≤(∫

vp φmR

)pq−1

≤ CRθ.

If α > n− 2, then θ < 0, and by letting R →∞ we immediately obtain v ≡ 0.Since uq ≤ −∆v = 0, we also get u ≡ 0.

If α = n−2, then θ = 0, so that (31.24) implies∫

vp < ∞. Returning to (31.24),we then deduce (∫

vp φmR

)pq

≤ C

∫R<|x|

vp → 0, as R →∞,

hence again v ≡ 0 and u ≡ 0.By exchanging the roles of u and v, we get the same conclusion if β ≥ n−2.

31.3. A priori bounds by the rescaling method

Unlike the method based on moving planes and Pohozaev-type identity, the rescal-ing method allows to treat more general systems of the form (31.1). However, onehas to assume, roughly speaking, that for each fixed x ∈ Ω, f and g behaveasymptotically like homogeneous functions of u, v. Several choices of homogeneityare possible. In this subsection, we shall work under the following assumptions:

f(x, u, v) = a(x)vp + f1(x, u), |f1| ≤ C(1 + ur), r <p(q + 1)p + 1

,

g(x, u, v) = b(x)uq + g1(x, v), |g1| ≤ C(1 + vs), s <q(p + 1)q + 1

,

a, b ∈ C(Ω), a, b > 0 in Ω, f1, g1 ∈ C(Ω× R).

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(31.25)

Theorem 31.13. Assume Ω bounded. For given p, q > 1, let (31.25) be satisfiedand assume that system (31.21) does not admit any bounded, positive classicalsolution. Then any nonnegative classical solution of (31.1) satisfies the a prioriestimate (31.7).

Theorem 31.13 is a variant of results from [180], [184] (see also referencestherein). Similarly as in the scalar case (cf. Corollary 10.3), existence results canbe deduced from Theorem 31.13 under suitable additional assumptions on f, g.

Proof. Let us first observe that, due to Remark 31.11(ii), the assumption of thetheorem guarantees that (31.22) neither has any nontrivial solution.

Similarly as in the proof of Theorem 12.1, we proceed by contradiction. Assumethat there exists a sequence (uj , vj) of solutions such that ‖uj‖∞ + ‖vj‖∞ → ∞.

264 III. Systems

We may assume ‖uj‖∞ ≥ ‖vj‖α/β∞ without loss of generality. Let xj ∈ Ω be such

that uj(xj) = ‖uj‖∞ and set

λj :=(‖uj‖1/α

∞ + ‖vj‖1/β∞)−1 → 0, as j →∞.

By passing to a subsequence, we may assume that xj → x∞ ∈ Ω. Setting dj :=dist(xj , ∂Ω), we then split the proof into two cases, according to whether dj/λj →∞ (along some subsequence) or dj/λj is bounded.

Case 1: dj/λj →∞. We rescale the solutions around xj as follows:

uj(y) = λαj uj(xj + λjy), vj(y) = λβ

j vj(xj + λjy), y ∈ Ωj ,

where Ωj = y ∈ Rn : |y| < dj/λj. Due to the definition of λj , it is clear that

uj(y), vj(y) ≤ 1, y ∈ Ωj . (31.26)

Moreover, u1/αj (0) = λj ‖uj‖1/α

∞ ≥ λj (‖uj‖1/α∞ + ‖vj‖1/β

∞)/2 = 1/2, hence

uj(0) ≥ 2−α. (31.27)

Now, since α + 2 = βp and β + 2 = αq, we find that (u, v) = (uj , vj) satisfiesthe system

−∆u = a(xj + λjy) vp + fj(y), y ∈ Ωj ,

−∆v = b(xj + λjy) uq + gj(y), y ∈ Ωj .

(31.28)

Here, fj(y) = λα+2j f1(xj +λjy, λ−α

j uj(y)) and gj(y) = λβ+2j g1(xj +λjy, λ−β

j vj(y)).In view of our assumption (31.25) with r < p(q + 1)/(p + 1) = (α + 2)/α, we have

|fj| ≤ Cλα+2j (1 + λ−αr

j )→ 0, as j →∞. (31.29)

Similarly we obtain|gj | → 0, as j →∞. (31.30)

For each fixed R > 0, we have B2R ⊂ Ωj for j sufficiently large, and |∆uj |, |∆vj |≤ C(R) in B2R, owing to (31.26), (31.28)–(31.30). It follows from interior ellipticLp-estimates that the sequences uj , vj are bounded in W 2,m(BR) for all 1 < m <

∞. By embedding theorems, we deduce that they are bounded in C1+γ(BR) foreach γ ∈ (0, 1). It follows that some subsequence of (uj , vj) converges, locallyuniformly on R

n, to a bounded nonnegative (classical) solution of

−∆U = a0 V p, y ∈ Rn,

−∆V = b0 U q, y ∈ Rn,

where a0 = a(x∞) > 0, b0 = b(x∞) > 0. Note that a0, b0 can easily be scaled outto be 1. But since U(0) ≥ 2−α due to (31.27), this contradicts the Liouville-typeproperty.

Case 2: dj/λj is bounded. We may assume that dj/λj → c ≥ 0. We perform thesame change of coordinates z = z(x) = (z1, z2, · · · , zn) as in Case 2 of the proof ofTheorem 12.1. Then the solution (u, v) = (uj(z), vj(z)) = (uj(x), vj(x)) satisfiesthe following system in a half ball:

−∑i,k

aik(z)∂2u

∂zi∂zk−∑

i

bi(z)∂u

∂zi= a(x(z))vp + f1(x(z), u), |z| < ε, z1 > 0,

−∑i,k

aik(z)∂2v

∂zi∂zk−∑

i

bi(z)∂v

∂zi= b(x(z))uq + g1(x(z), v), |z| < ε, z1 > 0,

u = v = 0, |z| < ε, z1 = 0.

Moreover, xj becomes zj := z(xj) = (dj , 0, 0, . . . , 0). Now we rescale (u, v) aroundzj by setting

uj(y) = λαj uj(zj + λjy), vj(y) = λβ

j vj(zj + λjy), y ∈ Ωj ,

with

Ωj =y :∣∣∣y− zj

λj

∣∣∣ < ε′

λj, y1 >

−dj

λj

and Σj =

y :∣∣∣y− zj

λj

∣∣∣ < ε′

λj, y1 =

−dj

λj

.

The rescaled system becomes

−∑i,k

aik(zj + λjy)∂2u

∂yi∂yk− λj

∑i

bi(zj + λjy)∂u

∂yi

= a(x(zj + λjy)) vp + fj(y), y ∈ Ωj ,

−∑i,k

aik(zj + λjy)∂2v

∂yi∂yk− λj

∑i

bi(zj + λjy)∂v

∂yi

= b(x(zj + λjy)) uq + gj(y), y ∈ Ωj ,

u = v = 0, y ∈ Σj ,

where

fj(y) = λα+2j f1(x(zj + λjy), λ−α

j uj(y)), gj(y) = λβ+2j g1(x(zj + λjy), λ−β

j vj(y)).

Passing to the limit, similarly as in Case 2 of the proof of Theorem 12.1, we endup with a nonnegative solution (U, V ) of

−∆U = a0 V p, y ∈ Rn, y1 > −c,

−∆V = b0 U q, y ∈ Rn, y1 > −c,

U = V = 0, y ∈ Rn, y1 = −c,

⎫⎪⎬⎪⎭

266 III. Systems

with U(0) ≥ 2−α. This yields a contradiction with the Liouville-type property ina half-space mentioned at the beginning of the proof.

31.4. A priori bounds by the Lpδ alternate bootstrap method

The method presented in this subsection relies on a specific bootstrap procedurein the scale of weighted Lebesgue spaces Lp

δ(Ω). A simpler bootstrap argumentalso relying on Lp

δ-spaces has been presented for the scalar case in Section 11.Unlike the moving planes or rescaling methods, the Lp

δ bootstrap method appliesto very weak solutions, and in particular it provides L∞-regularity results for suchsolutions. Also, it does not suppose any monotonicity or restricted dependence,nor scale invariance properties. On the other hand, it assumes stronger growthrestrictions than the previous two methods (for instance, for system (31.2) onehas to assume max(α, β) > n − 1 instead of α + β > n − 2). However, it willturn out that its growth conditions are optimal in the class of very weak solutions(see Theorem 31.16 below).

We consider general systems of the form (31.1), essentially under only an uppergrowth bound of the form

f(x, u, v) ≤ C1(1 + vp + ur),

g(x, u, v) ≤ C1(1 + uq + vs),

u, v ≥ 0, x ∈ Ω. (31.31)

We also assume a standard (mild) superlinearity condition:

f(x, u, v)+g(x, u, v) ≥ λ(u+v)−C1, u, v ≥ 0, x ∈ Ω, for some λ > λ1.(31.32)

Here f, g : Ω × [0,∞)2 → [0,∞) are Caratheodory functions, p, q > 1, r, s ≥ 1,C1 > 0. In what follows, we refer to the notion of L1

δ, or very weak, solutionintroduced in Definition 3.1. The following result is due to [449].

Theorem 31.14. Assume Ω bounded and (31.31), (31.32), with

max(α, β) > n− 1 (31.33)

andr, s < pBT =

n + 1n− 1

. (31.34)

Then any nonnegative very weak solution (u, v) of (31.1) belongs to L∞ × L∞(Ω)and satisfies the a priori estimate (31.7).

Similarly as in the scalar case (cf. Corollary 10.3), existence results can bededuced from Theorem 31.14 under suitable additional assumptions on f, g. Con-dition (31.32) can be weakened or replaced by other conditions of different form.

For instance, by applying the same method, we obtain regularity and a prioriestimate for the following simple system:

−∆u = a(x)vp, x ∈ Ω,

−∆v = b(x)uq, x ∈ Ω,

u = v = 0, x ∈ ∂Ω.

⎫⎪⎬⎪⎭ (31.35)

Theorem 31.15. Assume Ω bounded, p, q > 1, a, b ∈ L∞(Ω), a, b ≥ 0, a, b ≡0 and (31.33). Then any nonnegative very weak solution of (31.35) belongs toL∞ × L∞(Ω) and satisfies the a priori estimate (31.7). Moreover, there exists asolution (u, v) of (31.35), with u, v ∈ C0 ∩W 2,m(Ω) for all finite m, and u, v > 0.

Theorem 31.15 is from [489] (see also [449]). The optimality of condition (31.33)in Theorems 31.14 and 31.15 is shown by the following result from [489], whichwill be proved at the end of this section (see Theorem 11.5 for the analogue in thescalar case).

Theorem 31.16. Assume Ω bounded, p, q > 1 and

max(α, β) < n− 1. (31.36)

Then there exist functions a, b ∈ L∞(Ω), a, b ≥ 0, a, b ≡ 0, such that system(31.35) admits a positive very weak solution (u, v) satisfying

u ∈ L∞(Ω), v ∈ L∞(Ω).

In the bootstrap procedure in the proof of Theorems 31.14 and 31.15, eachequation is used alternatively. At each step, we make use of the Lp

δ regularitytheory (cf. Theorem 49.2 and Proposition 49.5 in Appendix C), and L∞ is reachedafter finitely many steps. The proof of Theorem 31.15 given below presents thesimplest case of application of these ideas to systems. The proof of Theorem 31.14,although based on the same basic approach, is more involved and will not be givenhere.

Proof of Theorem 31.15. Step 1. Initialization. By testing with ϕ1, we obtainthe basic estimate

∫Ω u dxϕ1,

∫Ω v dxϕ1 ≤ C, i.e.

‖u‖1,δ + ‖v‖1,δ ≤ C (31.37)

in view of (1.4). (In the case a(x), b(x) ≥ C > 0, this is Step 1 of the proof ofTheorem 31.2. For general a, b, this can be done by a simple modification usingthe argument in the proof of Theorem 11.3.)

We set f := a(x)vp and g := b(x)uq. Then (31.37) guarantees ‖f‖1,δ + ‖g‖1,δ ≤C. Assume without loss of generality q ≥ p and β = 2(q+1)

pq−1 > n− 1. In particular,

there holds (p− 1)(q + 1) ≤ pq − 1 < 2(q+1)n−1 hence

p < pBT . (31.38)

268 III. Systems

Proposition 49.5 guarantees that

‖u‖k,δ + ‖v‖k,δ ≤ C(k), for all 1 ≤ k < pBT . (31.39)

Note that if n = 1, then the growth assumptions on f, g and Theorem 49.2(i)immediately imply ‖u‖∞ + ‖v‖∞ ≤ C. We may thus assume n ≥ 2.

We will show by a bootstrap argument that the value of k in (31.39) can beincreased so as to reach k = ∞. Thus assume that there holds

‖u‖k,δ + ‖v‖k,δ ≤ C(k) (31.40)

for some k satisfyingk ≥ p and k ≥ pBT − ε, (31.41)

where ε = ε(p, q, n) > 0 small will be chosen below.Step 2. Bootstrap on the first equation. Let k1 ∈ (k,∞] satisfy

1k1

>p

k− 2

n + 1. (31.42)

Using Theorem 49.2(i) and the first equation, we obtain

‖u‖k1,δ ≤ C‖∆u‖k/p,δ = C‖f‖k/p,δ ≤ C‖vp‖k/p,δ = C‖v‖pk,δ ≤ C. (31.43)

For later use, we already note that if

k >(n + 1)pq

2(q + 1), (31.44)

then by taking ε = ε(n, p) > 0 in (31.41) sufficiently small, we may find

k1 >(n + 1)q

2(31.45)

such that (31.43) is satisfied. Indeed, pk −

2n+1 < min

(2

(n+1)q , 1k

)and we may thus

find k1 ∈ (k,∞) satisfying (31.45) and (31.42), hence (31.43).Step 3. Bootstrap on the second equation. Now assume

k1 > q (31.46)

and let k2 ∈ (k,∞] satisfy1k2

>q

k1− 2

n + 1. (31.47)

Using Theorem 49.2(i), the second equation and (31.43), we obtain

‖v‖k2,δ ≤ C‖∆v‖k1/q,δ = C‖g‖k1/q,δ ≤ C‖uq‖k1/q,δ = C‖u‖qk1,δ ≤ C. (31.48)

Step 4. Fulfillment of the bootstrap conditions. Let ρ = ρ(p, q, n) ∈ (0, 1) tobe determined. Conditions (31.42), (31.46), (31.47), together with the bootstrapcondition

min(k1, k2) >k

ρ,

are equivalent to

A :=p

k− 2

n + 1<

1k1

< min(ρ

k,1q

)(31.49)

andq

k1− 2

n + 1<

1k2

<ρ

k. (31.50)

Assume

k ≤ (n + 1)pq

2(q + 1)(31.51)

hence, in particular, A > 0. Then condition (31.49) can be solved in k1 ∈ [1,∞),and 1/k1 can be taken arbitrarily close to A, provided

p− ρ

k<

2n + 1

(31.52)

andp

k− 2

n + 1<

1q. (31.53)

Since k ≥ p, condition (31.52) is satisfied whenever

n− 1n + 1

p < ρ < 1, (31.54)

which is allowable in view of (31.38). Due to β > n− 1, we have (pq− 1)(n− 1) <2q + 2 hence n−1

n+1p− 2n+1 < 1

q . Taking ε = ε(p, q, n) > 0 small in (31.41), we thusget (31.53).

On the other hand, condition (31.50) can be solved in k2 ∈ [1,∞) if

q

k1− 2

n + 1<

ρ

k. (31.55)

Taking 1/k1 in (31.49) close enough to its lower bound A (cf. after (31.51)), (31.55)becomes equivalent to

ρ > 1− η, where η :=2(q + 1)n + 1

k − (pq − 1). (31.56)

Observe that η > 0 is equivalent to k > (n+1)/β and, since β > n−1, this is truefor ε = ε(p, q, n) > 0 small in (31.41). We may thus choose ρ = ρ(p, q, n) ∈ (0, 1)satisfying (31.54) and (31.56).

270 III. Systems

Step 5. Conclusion. We deduce from Step 4 that if (31.40) holds for some ksatisfying (31.41) and (31.51), then (31.40) is true with k replaced by k/ρ. Startingfrom (31.39), we see that some value k > (n + 1)pq/2(q + 1) of k is reached aftera finite number of steps. It then follows from the second paragraph in Step 2 that‖u‖k1,δ ≤ C for some k1 > (n + 1)q/2 ≥ (n + 1)p/2.

By Step 3 with k1 := k1 and k2 := ∞, it follows that ‖v‖∞ ≤ C. We may thenapply Step 2 with k := k1 and k1 := ∞ to conclude that ‖u‖∞ ≤ C. The proof iscomplete.

As an application of the methods in this section, one obtains the following result[449] concerning the system

−∆u = uv − au, x ∈ Ω,

−∆v = bu, x ∈ Ω,

u = v = 0, x ∈ ∂Ω,

⎫⎪⎬⎪⎭ (31.57)

mentioned in Remark 31.1(i).

Theorem 31.17. Assume Ω bounded, a, b > 0, and n ≤ 4. Then any nonnegativevery weak solution of (31.57) belongs to L∞ × L∞(Ω) and satisfies the a prioriestimate (31.7). Moreover, there exists a classical solution of (31.57) with u, v > 0.

Sketch of proof (see [449] for details). We use a variant of Theorem 31.14. Infact, without assuming (31.32), the growth conditions (31.31), (31.33), (31.34)alone ensure that any very weak solution satisfies u, v ∈ L∞ ∩W 2,m(Ω) for allfinite m. Moreover, if we know an a priori estimate of u and v in L1

δ(Ω), thenthis implies an a priori estimate in L∞(Ω) (the only role of assumption (31.32) inTheorem 31.14 is to guarantee the L1

δ-estimate).Take 1 < r < pBT , p = r/(r − 1) and q = 1. Using uv ≤ vp + ur, and noting

that max(α, β) = 2(p + 1)/(pq − 1) = 4r − 2 > n − 1 for r close to pBT due ton < 5, we see that f = uv − au, g = bu satisfy (31.31), (31.33), (31.34).

On the other hand, the L1δ a priori estimate can be shown as follows. We have

−∆u = −b−1v∆v − au ≥ −b−1

2∆(v2)− au.

Testing this inequality and the second equation in (31.57) with ϕ1, we obtain

(λ1 + a)∫

Ω

uϕ1 dx ≥ b−1λ1

2

∫Ω

v2ϕ1 dx ≥ b−1λ1

2

(∫Ω

vϕ1 dx)2

=λ−1

1 b

2

(∫Ω

uϕ1 dx)2

.

This implies the desired estimate.

Remarks 31.18. Comparison with other methods. (i) The method of Sec-tion 10 based on Hardy-Sobolev inequalities has also been extended to certainsystems, see [141], [258], [135], [144]. Like the Lp

δ bootstrap method, it essentiallyrequires only upper bounds on the growth of the nonlinearities f, g. However, thegrowth restrictions on the nonlinearities are much stronger, unlike in the scalarcase (roughly, min(α, β) > n − 1 instead of max(α, β) > n − 1; cf. [135]). Thereason for this is that the bootstrap procedure in that method is based on anH1×H1-estimate and is carried out simultaneously on the two components. Con-sequently, unlike in the above proof, the possible compensation effects between thetwo equations are not fully exploited.

(ii) Condition (31.33) also appears in the work [85], where existence and a prioriestimates are studied for system (31.2) with extra (measure) terms added in theRHS and in the boundary conditions. The method in [85] is different from thatdescribed in this section. In particular, it uses maximum principle arguments toderive comparison estimates of the form uq+1 ≤ C(1+vp+1). In the case of system(31.1) (without measures in the RHS), it applies typically when 0 ≤ f ≤ C2v

p andC1u

q ≤ g ≤ C2uq, with C2 ≥ C1 > 0 and p, q satisfying (31.33).

We now turn to the proof of Theorem 31.16. Like that of Theorem 11.5, itis mainly a consequence of Lemma 49.13, where a singular solution of the linearLaplace equation with an appropriate right-hand side belonging to L1

δ is con-structed.

Proof of Theorem 31.16. Set φ := |x|−(α+2)χΣ and ψ := |x|−(β+2)χΣ, withΣ as in Lemma 49.13 and let u, v > 0 be the very weak solutions of (47.8) withf = φ, ψ, respectively. By (49.29), we have u ∈ L∞, v ∈ L∞ and

vp ≥ C|x|−βpχΣ = C|x|−(α+2)χΣ = Cφ,

uq ≥ C′|x|−αqχΣ = C′|x|−(β+2)χΣ = C′ψ.

Setting a(x) = φ/vp ≥ 0, b(x) = ψ/uq ≥ 0, we get−∆u = φ = a(x)vp,−∆v = ψ =b(x)uq and a(x) ≤ 1/C, b(x) ≤ 1/C ′ hence a, b ∈ L∞. The proof is complete.

Remark 31.19. Localization of singularities. The observations in Remarks11.6 extend to the case of systems. In particular, in spite of the imposed homo-geneous Dirichlet boundary condition, the singularities of the solution in Theo-rem 31.15 occur at a (single) boundary point. In fact, when n− 2 < max(α, β) <n− 1, system (31.1) cannot have purely interior singularities. On the contrary, formax(α, β) < n − 2, examples of similar systems which possess unbounded weaksolutions with purely interior singularities can be easily constructed. Namely thepair (u, v) = (r−α − 1, r−β − 1), r = |x|, is a weak solution of system (31.1) withf = c1(v + 1)p and g = c2(u + 1)q for Ω = B1 and suitable constants c1, c2 > 0(note that the right-hand sides are in L1).

272 III. Systems

32. Parabolic systems coupled by power sourceterms

In this section, as a simple superlinear parabolic system and an analogue of thescalar model problem (15.1), we study the system:

ut −∆u = |v|p−1v, x ∈ Ω, t > 0,

vt −∆v = |u|q−1u, x ∈ Ω, t > 0,

u = v = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

v(x, 0) = v0(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(32.1)

where p, q > 0. We set

X = L∞ × L∞(Ω) and X+ = (u0, v0) ∈ X : u0, v0 ≥ 0. (32.2)

In all this section, when pq > 1, the scaling exponents α, β are defined by (31.3).

Assume p, q ≥ 1. Then problem (32.1) is locally well-posed in X (see Exam-ple 51.12). In particular,

if Tmax < ∞, then limt→Tmax

(‖u(t)‖∞ + ‖v(t)‖∞

)=∞. (32.3)

Also the solution satisfies

u, v ∈ BC2,1(Ω× [t1, t2]), 0 < t1 < t2 < Tmax. (32.4)

Furthermore, problem (32.1) admits a comparison principle (cf. Proposition 52.22).

Next consider the case p, q > 0 and min(p, q) < 1. For (u0, v0) ∈ X , localexistence can be proved easily by approximation arguments (similar to those inthe proof of Proposition 51.16 for instance). Turning to the question of (non-)uniqueness, which has been studied in [171], [172], let us assume (u0, v0) ∈ X+,and Ω bounded or Ω = R

n. Local uniqueness is true in the class of nonnegativeclassical solutions if either pq ≥ 1 or (u0, v0) = (0, 0), but the proof is nontrivial. Onthe contrary, there exist infinitely many nonnegative classical solutions if pq < 1and (u0, v0) = (0, 0). On the other hand, if p, q > 0 and (u, v) is any maximalclassical solution of (32.1) with existence time denoted by Tmax, then we still have(32.3) and (32.4).

32. Parabolic systems coupled by power source terms 273

32.1. Well-posedness and continuation in Lebesgue spaces

We consider system (32.1) with initial values in the space Y = Lr1 × Lr2(Ω). For(u0, v0) ∈ Y , by a local solution of (32.1) (on [0, T ]), we understand a function(u, v) ∈ C([0, T ], Y ) which is a classical solution of (32.1) for 0 < t ≤ T and whichfulfills the initial conditions. (Actually the nonexistence result below will still holdfor a weaker notion of solution, see [446] for details.)

The optimal condition for local existence/nonexistence for system (32.1) can beexpressed in terms of the numbers

P = n( p

r2− 1

r1

), Q = n

( q

r1− 1

r2

).

Theorem 32.1. (i) (Well-posedness) Let p, q > 1, r1, r2 > 1 and assume

max(P ,Q) ≤ 2.

For all (u0, v0) ∈ Lr1 × Lr2(Ω), there exist T > 0 and a unique local solution ofsystem (32.1) on [0, T ].(ii) (Local nonexistence) Let p, q > 0, r1, r2 ≥ 1 and assume

max(P ,Q) > 2.

Then there exists (u0, v0) ∈ Lr1 × Lr2(Ω), u0, v0 ≥ 0, such that system (32.1)admits no local solution (u, v) with u, v ≥ 0.

As in Section 16, it is natural to look for sufficient conditions, in terms of Lr-bounds, guaranteeing global existence.

Theorem 32.2. (Continuation) Let p, q ≥ 1, pq > 1, n ≥ 2 and assume Ωbounded. Let (u, v) be a maximal classical solution of (32.1) and denote by T itsexistence time. Assume that either

r1 >n

α=

n(pq − 1)2(p + 1)

and sup(0,T )

‖u(t)‖r1 < ∞,

or

r2 >n

β=

n(pq − 1)2(q + 1)

and sup(0,T )

‖v(t)‖r2 < ∞.

Then T =∞.

Theorems 32.1 and 32.2 are from [446] and [447], respectively. Observe that theinequality max(P ,Q) < 2 implies r1 > n/α and r2 > n/β, but that this can betrue also when max(P ,Q) > 2. Therefore, the continuation property is valid underweaker assumptions on r1, r2 than well-posedness. This is in sharp contrast withthe situation in the scalar case (cf. Theorems 15.2 and 15.3, and Corollary 24.2).Note also that an Lr-bound on a single component is enough to guarantee globalexistence.

274 III. Systems

Remark 32.3. The gap between conditions guaranteeing well-posedness and con-tinuation can be heuristically explained as follows. The final profiles of a solutionaround a blow-up point x0 are expected to verify the lower estimates

u(x, T ) ≥ c1|x− x0|−α, v(x, T ) ≥ c2|x− x0|−β

for |x−x0| small (cf. Remark 32.12(ii) for a partial result), hence (u(·, T ), v(·, T )) /∈Lr1 × Lr2 whenever r1 > n/α or r2 > n/β. On the other hand, if a local solutionexists, then u0 and v0 have to satisfy suitable integral estimates as a consequenceof the variation-of-constants formula (see (32.8) below), and this leads to necessaryconditions involving r1 and r2 if (32.1) is well-posed in Lr1 × Lr2 .

Theorem 32.1(i) is proved in Example 51.32 of Appendix E. As for the proof ofTheorem 32.1(ii), the main ingredient is the following lemma, which provides lowerestimates for certain time-space averages of solutions of the linear heat equationwith positive singular initial data.

Lemma 32.4. Assume 0 2.

Then there exists v0 ∈ Lr2(Ω), v0 ≥ 0, such that

∥∥∥∫ t

0

e−(t−s)A(e−sAv0

)pds∥∥∥

r1

→∞, as t → 0+.

Proof. Assume B(0, 2ρ) ⊂ Ω, ρ > 0, let k ∈ (0, n/r2), and define

v0(y) = |y|−kχB(0,ρ)(y).

Clearly, v0 ∈ Lr2(Ω). Using the heat kernel estimate in Proposition 49.10, weobtain, for s > 0 small and |x| ≤

√s/2,

(e−sAv0

)(x) =

∫|y|<ρ

G(x, y, s)|y|−k dy ≥ c1s−n/2

∫|y−x|<√

s|y|−k dy

≥ c1s−n/2

∫|y|<√

s/2|y|−k dy ≥ cs−k/2.

Consequently,

e−sAv0 ≥ cs−k/2 χB(0,√

s/2), for s > 0 small. (32.5)

Next, let t/4 ≤ s ≤ t/2, with t > 0 small, and |x| ≤√

s/2. For |y| ≤√

s/2we have |x − y| ≤

√t− s, hence G(x, y, t − s) ≥ c1(t − s)−n/2 ≥ c1s

−n/2 byProposition 49.10. It follows that(

e−(t−s)AχB(0,√

s/2)

)(x) =

∫|y|<√

s/2

G(x, y, t− s) dy ≥ c > 0.

Combining this with (32.5), we deduce that, for t > 0 small,


)p(x) ≥ cs−kp/2 ≥ ct−kp/2, t/4 ≤ s ≤ t/2, |x| ≤√

s/2.(32.6)

Now if |x| ≤√

t/4 and t is small, it follows from (32.6) that(∫ t

0


)pds)(x) ≥

(∫ t/2

t/4


)pds)(x) ≥ Ct1−

kp2

(note that s ≥ t/4 implies√

s/2 ≥√

t/4 ≥ |x|). Therefore, for t > 0 small, weobtain∥∥∥∫ t

0


)pds∥∥∥r1

r1

≥∫|x|≤√

t/4

(∫ t

0


)pds)r1

(x) dx ≥ Ctn2 +r1(1− kp

2 ).

Since the assumption of the lemma implies n2 + r1

(1 − np

2r2

)< 0, the conclusion

follows by choosing k sufficiently close to n/r2.

Proof of Theorem 32.1(ii). Similarly as in Remark 15.4(iii), any nonnegativesolution of (32.1) in the sense of Theorem 32.1 satisfies the variation-of-constantsformula:

u(t) = e−tAu0 +∫ t

0

e−(t−s)A|v(s)|p−1v(s) ds, 0 ≤ t < T,

v(t) = e−tAv0 +∫ t

0

e−(t−s)A|u(s)|q−1u(s) ds, 0 ≤ t < T.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (32.7)

In particular, we have

u(t) ≥ e−tAu0 ≥ 0, 0 ≤ t < T,

v(t) ≥ e−tAv0 ≥ 0, 0 ≤ t < T.

It follows that

0 ≤∫ t

0

e−(t−s)A(e−sAv0)p ds ≤ u(t),

0 ≤∫ t

0

e−(t−s)A(e−sAu0)q ds ≤ v(t).

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (32.8)

276 III. Systems

Since (u, v) ∈ C([0, T ], Lr1 × Lr2), the right-hand sides in (32.8) remain boundedin Lr1 or Lr2 , respectively, hence∥∥∥∫ t

0


)pds∥∥∥

r1

+∥∥∥∫ t

0

e−(t−s)A(e−sAu0

)qds∥∥∥

r2

≤ C, 0 < t < T.

If either P > 2 or Q > 2, that is,

n( p

r2− 1

r1

)> 2 or n

( q

r1− 1

r2

)> 2,

then, by choosing u0 ∈ Lr1 or v0 ∈ Lr2 as given by Lemma 32.4, we conclude thatno solution of (32.1) can exist.

Proof of Theorem 32.2. We shall prove the result only for n ≥ 4. The prooffor n = 2, 3 is more involved and relies on suitable interpolation spaces. (However,the proof below applies also if n = 3 and p, q ≥ 2, or if n ≤ 3 and r1 > q − (1/p),r2 > p− (1/q).)

By Propositions 48.4 and 48.5, there exists ω > 0 such that

‖e−tA‖L(Lm1 ,Lm2) ≤ C1t−n

2 ( 1m1

− 1m2

)e−ωt, 1 ≤ m1 ≤ m2 ≤ ∞. (32.9)

By a time shift, we may assume that (u, v) is smooth up to t = 0. In particular, itsatisfies the variation-of-constants formula (32.7).

We fix τ ∈ (0, T ) and we denote

|u|m := supt∈(0,τ)

‖u(t)‖m < ∞, 1 ≤ m ≤ ∞

(and similarly for v). In the rest of the proof, C denotes a generic constant inde-pendent of τ . Assume that

|v|r ≤ C (32.10)

for some

r >n(pq − 1)2(q + 1)

. (32.11)

Let k, l satisfy

1 ≤ k ≤ l < ∞, k ≥ r

pand

1k− 1

l<

2n

.

By the first equation in (32.7) and the smoothing property (32.9) with m1 = k,m2 = l it follows that

|u|l ≤ C(1 +∣∣|v|p∣∣

k) = C(1 + |v|pkp)

hence, by (32.10) and by the interpolation inequality,

|u|l ≤ C(1 + |v|p−(r/k)∞ ). (32.12)

If in additionl >

nq

2,

then the second equation in (32.7) and (32.9) with m1 = l/q, m2 = ∞ imply|v|∞ ≤ C

(1 +

∣∣|u|q∣∣l/q

)= C(1 + |u|ql );

hence, by (32.12),|v|∞ ≤ C(1 + |v|q(p−(r/k))

∞ ).It follows that |v|∞ ≤ C if

pq − 1qr

<1k

.

The sufficient conditions are thus

max(0,

1k− 2

n

)<

1l

< min( 2

nq,1k

)(32.13)

andpq − 1

qr<

1k

< min(1,

p

r

). (32.14)

Condition (32.13) can be solved in l if1k− 2

n<

2nq

,

i.e.,1k

<2(q + 1)

nq.

Sincepq − 1

qrn(pq − 1)2(q + 1)

and r > p− 1q.

Finally, note thatn(pq − 1)2(q + 1)

≥ p− 1q

if (n− 2)q ≥ 2,

which is true for all q ≥ 1 if n ≥ 4, and for q ≥ 2 if n = 3. The hypothesis(32.11) thus implies the solvability of (32.13)–(32.14). Consequently, |v|∞ ≤ C,hence |u|∞ ≤ C by the first equation in (32.7). Since C is independent of τ , wededuce that u and v are uniformly bounded on QT , hence T = ∞.

278 III. Systems

32.2. Blow-up and global existence

The following result provides the conditions on the exponents p, q which imply orprevent blow-up for system (32.1) in bounded domains.

Theorem 32.5. Assume Ω bounded, p, q > 0, (u0, v0) ∈ X+, and set p =min(p, 1), q = min(q, 1). Let (u, v) be a maximal classical solution of (32.1) anddenote by T its existence time.(i) If pq > 1, then there exists C(p, q, Ω) > 0 with the following property: If∫Ω (uq

0 + vp0)ϕ1 dx > C(p, q, Ω), then T < ∞.

(ii) If pq ≤ 1, then T = ∞. Moreover, if pq < 1, then u(t), v(t) are uniformlybounded for t ≥ 0.

Theorem 32.5 is a modification of a result from [173] (see also [224], [226] forp, q > 1).

Proof. (i) Denote y = y(t) :=∫Ω

u(t)ϕ1 dx, z = z(t) :=∫Ω

v(t)ϕ1 dx. We mayassume q = max(p, q) > 1 without loss of generality. Multiplying the secondequation in (32.1) with ϕ1, we have

z′ =∫

Ω

vtϕ1 dx =∫

Ω

v∆ϕ1 dx +∫

Ω

uqϕ1 dx.

Using ∆ϕ1 = −λ1ϕ1 and Jensen’s inequality yields

z′ ≥ −λ1z + yq. (32.15)

We first consider the easier case p > 1. Similarly as above, we have

y′ ≥ −λ1y + zp.

Therefore φ := y + z satisfies

φ′ = y′ + z′ ≥ −λ1φ + zp + yq ≥ −λ1φ + zp + yp − y ≥ −(1 + λ1)φ + 21−pφp.

It follows that T <∞ whenever φ(0) > C(λ1, p).Next consider the case p ≤ 1. In what follows, the constants ci > 0 will depend

only on p, q, Ω. Recall that (u, v) satisfies the variation-of-constants formula (32.7).By (15.24), for each 0 < σ < s < t, we have e−(s−σ)Aupq(σ) ≤

(e−(s−σ)Auq(σ)

)p,hence ∫ s

0

e−(s−σ)Aupq(σ) dσ ≤∫ s

0

(e−(s−σ)Auq(σ)

)pdσ

≤ s1−p(∫ s

0

e−(s−σ)Auq(σ) dσ)p

.

Using (32.7), we deduce that

u(t) ≥ e−tAu0 +∫ t

0

e−(t−s)A(∫ s

0

e−(s−σ)Auq(σ) dσ)p

ds

≥ e−tAu0 + tp−1

∫ t

0

e−(t−s)A(∫ s

0

e−(s−σ)Aupq(σ) dσ)

ds,

hence

u(t) ≥ e−tAu0 + tp−1

∫ t

0

∫ s

0

e−(t−σ)Aupq(σ) dσ ds (32.16)

by Fubini’s theorem. Put γ = pq > 1. It follows from Jensen’s inequality that

(e−(t−σ)Auγ(σ), ϕ1) = e−λ1(t−σ)(uγ(σ), ϕ1) ≥ e−λ1(t−σ)yγ(σ).

Multiplying (32.16) with ϕ1, we thus obtain

y(t) ≥ e−λ1ty(0) + tp−1

∫ t

0

∫ s

0

e−λ1(t−σ)yγ(σ) dσ ds.

Assume that T ≥ 1. We have

y(t) ≥ c1y(0) + c1

∫ t

0

∫ s

0

yγ(σ) dσ ds =: h(t), 0 < t < 1,

henceh′′(t) ≥ c1h

γ , 0 < t < 1. (32.17)

To conclude, it suffices to show that this inequality cannot be satisfied whenever∫Ω

(u0 + vp0)ϕ1 dx ≥ M, where M = M(p, q, Ω) is large enough.

Multiplying (32.17) by h′ ≥ 0 and integrating, we have

h′2(t) ≥ c2hγ+1(t)− c3y

γ+1(0), 0 < t < 1. (32.18)

On the other hand, using (32.7) and p ≤ 1 again, we get

z(t) :=∫

Ω

vp(t)ϕ1 dx ≥∫

Ω

(e−tAv0)pϕ1 dx ≥∫

Ω

(e−tAvp0)ϕ1 dx = e−λ1tz(0)

and next,

y(t) = e−λ1ty(0) +∫ t

0

e−λ1(t−s)

∫Ω

vp(s)ϕ1 dx ds ≥ c4(y(0) + t z(0)), 0 < t < 1.

280 III. Systems

Therefore, since h′′(t) = c1yγ , we have

h(1/2) ≥ c5(y(0) + z(0))γ . (32.19)

Due to γ > 1, if y(0) + z(0) ≥ M (where M is large enough), we deduce from(32.18) that h′ ≥ c6h

(γ+1)/2 on (1/2, 1), which contradicts (32.19) for M large.(ii) Let us first assume pq < 1. Let Θ be the classical solution of (19.27), and

put M = ‖Θ‖∞. We observe that (u, v) = (a(1 + Θ), b(1 + Θ)) is a supersolution,whenever the constants a, b > 0 satisfy a ≥ [b(1 + M)]p and b ≥ [a(1 + M)]q. It isthus sufficient that (1 + M)pbp ≤ a ≤ (1 + M)−1b1/q. Since p < 1/q, for a given(u0, v0), one can take a, b as above and such that a ≥ ‖u0‖∞, b ≥ ‖u0‖∞. Theassertion then follows from the comparison principle (note that since u ≥ a > 0and v ≥ b > 0, it applies even though p, q may be < 1 — see Remark 52.11(c)).

Finally assume pq = 1, and p ≥ 1 without loss of generality. We claim thatfor all a > 0, (u, v) = (apept, aet) is a supersolution. Indeed, this is equivalent topapept ≥ apept and aet ≥ apqepqt, which is true due to pq = 1 and p ≥ 1. It thensuffices to choose a ≥ max

(‖u0‖1/p

∞ , ‖v0‖∞).

Remark 32.6. Results on a priori estimates and universal bounds for globalpositive solutions of (32.1) can be found in [448] (see also [425, Section 6]).

32.3. Fujita-type results

In this subsection we consider nonnegative solutions of the Cauchy problem asso-ciated with (32.1), i.e.:

ut −∆u = vp, x ∈ Rn, t > 0,

vt −∆v = uq, x ∈ Rn, t > 0,

u(x, 0) = u0(x), x ∈ Rn,

v(x, 0) = v0(x), x ∈ Rn.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (32.20)

We give a Fujita-type result for problem (32.20), i.e. we find the (optimal) condi-tions on p, q depending on n, which guarantee that the solution blows up in finitetime for all u0, v0 ≥ 0, (u0, v0) ≡ (0, 0).

Theorem 32.7. Let p, q > 0 satisfy pq > 1, and let (u0, v0) ∈ X+, (u0, v0) ≡(0, 0).(i) If max(α, β) ≥ n, then (32.20) admits no nontrivial global solution.(ii) If max(α, β) < n, then (32.20) admits global, bounded solutions for suitablysmall initial data.

This result is from [171]. We will prove it only under the additional assumptionp, q ≥ 1, and we will not treat the critical value max(α, β) = n. However in


this special case, the present proof, based on arguments from [192], is considerablysimpler than that in [171]. We shall use Gaussian test-functions of x and differentialinequalities. See [372] for a different proof in the case p, q ≥ 1, based on rescaledtest-functions of x and t.

As a preliminary to the proof, we prepare the following lemma concerning thesystem of differential inequalities:

y′(t) ≥ zp − λy, t ≥ 0,

z′(t) ≥ yq − λz, t ≥ 0.

(32.21)

Lemma 32.8. Let p, q > 0 satisfy pq > 1 and λ > 0. Then there exists K =K(p, q) > 0 such that (32.21) has no global nonnegative solution y, z ∈ C([0,∞))∩C1((0,∞)) with y(0) ≥ Kλα/2.

Proof. Put τ = λ−1 and assume that (y, z) exists on [0, τ ]. Then there existsC1 = C1(q) > 0 such that

y(τ) ≥ C1y(0) and z(τ) ≥ C1λ−1yq(0). (32.22)

Indeed, we have (yeλt)′ ≥ 0, hence y(t) ≥ y(0)e−λt ≥ e−1y(0) on [0, τ ]. Thisimplies (zeλt)′ ≥ eλtyq(t) ≥ e−qyq(0) on [0, τ ], hence z(τ) ≥ e−(q+1)λ−1yq(0), and(32.22) follows.

Next, since pq > 1, we may choose A, B > 1 depending only on p, q, such thatp(B− 1) > A and q(A− 1) > B. We claim that if, for some t0, there exist a, b > 0such that

y(t0) > a, z(t0) > b, bp > Aλa, and aq > Bλb, (32.23)

then (y(t), z(t)) cannot exist globally.To prove the claim, assume for contradiction that (y, z) exists for all t > 0. By a

time shift, we may assume t0 = 0. Let (y, z) be the unique, positive local solutionof

y′(t) = zp − λy, t ≥ 0,

z′(t) = yq − λz, t ≥ 0,

y(0) = a, z(0) = b.

⎫⎪⎬⎪⎭By an easy comparison argument (using the fact that z → zp and y → yq areincreasing functions), it follows that (y, z) exists for all t > 0 and we have y(t) ≥y(t) > 0 and z(t) ≥ z(t) > 0. Set φ(t) = zp −Aλy and ψ(t) = yq −Bλz. We haveφ(0) > 0 and ψ(0) > 0 by (32.23). Assume that φ, ψ > 0 on [0, T ] for some T > 0.Then y′ ≥ (A − 1)λy and z′ ≥ (B − 1)λz on (0, T ]. On the other hand, for allt ∈ (0, T ], we have

φ′(t) = pzp−1z′ −Aλy′ ≥ (p(B − 1)−A)λzp > 0

282 III. Systems

andψ′(t) = qyq−1y′ −Bλz′ ≥ (q(A− 1)−B)λyq > 0.

We deduce that φ, ψ > 0 on [0,∞). It follows that y′(t) ≥ czp and z′(t) ≥ cyq

with c = 1 −max(A−1, B−1) > 0. But, as a consequence of Lemma 32.10 below,this guarantees that (y, z) cannot exist for all t > 0. This contradiction proves theclaim.

Let us now show that, for suitable K, ε, η > 0 (independent of λ), y(0) ≥ Kλα/2

guarantees that a := ελα/2 and b := ηλβ/2 satisfy (32.23) for t0 = τ . In view ofthe last claim, this will prove the lemma. The last two conditions in (32.23) areequivalent to

ηpλp(q+1)pq−1 > Aλελ

p+1pq−1 = Aελ

p(q+1)pq−1 , εqλ

q(p+1)pq−1 > Bληλ

q+1pq−1 = Bηλ

q(p+1)pq−1 ,

that is ηp > Aε and εq > Bη; such η, ε > 0 clearly exist since pq > 1. Due to(32.22), the first two conditions in (32.23) are satisfied if

ελp+1

pq−1 < C1Kλp+1

pq−1 , ηλq+1

pq−1 < C1λ−1Kqλ

q(p+1)pq−1 = C1K

qλq+1

pq−1 .

It thus suffices to choose K > max(C−11 ε, C

−1/q1 η1/q).

Proof of Theorem 32.7. (i) Without loss of generality, we may assume p ≥ q.As mentioned before, we shall prove the assertion under the stronger assumptionsp ≥ q ≥ 1 (p > 1) and max(α, β) = α > n. For each λ > 0, let φλ(x) =(4π)−n/2λn/2e−λ|x|2/4. We have

∂xiφλ =−λxi

2φλ, ∂2

xixiφλ =

[λ2x2i

4− λ

2

]φλ, hence ∆φλ ≥

−nλ

2φλ,

and∫

Rn φλ = 1. Multiplying the differential equations in (32.1) by φλ, integratingby parts, and using Jensen’s inequality, we obtain

d

dt

∫Rn

uφλ dx =∫

Rn

u∆φλ dx +∫

Rn

vpφλ dx ≥ −nλ

2

∫Rn

uφλ dx +(∫

Rn

vφλ dx)p

and similarly

d

dt

∫Rn

vφλ dx ≥ −nλ

2

∫Rn

vφλ dx +(∫

Rn

uφλ dx)q

.

(the calculations can be justified similarly as in the proof of Theorem 17.1). There-fore, the functions

yλ(t) :=∫

Rn

u(t)φλ dx and zλ(t) :=∫

Rn

v(t)φλ dx

satisfy system (32.21) with λ replaced by λ := nλ/2. By shifting the time origin, wemay assume u0 ≡ 0. Consequently, since

∫Rn e−λ|x|2/4u0 dx →

∫Rn u0 dx ∈ (0,∞]

as λ → 0, there exists c0 > 0 such that yλ(0) ≥ c0λn/2 for λ > 0 small. Since

α > n, we have yλ(0) ≥ Kλα/2 for λ > 0 small, where K is given by Lemma 32.8.We then deduce from that lemma that (yλ, zλ), hence (u, v), cannot exist globally.

(ii) We assume p ≥ q ≥ 1, p > 1 and α < n. We look for a supersolution underthe form

u(x, t) = ε(t + 1)aφ(x, t), v(x, t) = ε(t + 1)bφ(x, t),

with a, b, ε > 0 and φ(x, t) = (t + 1)−n/2ψ(x, t), where ψ(x, t) = e−|x|2/4(t+1).Using φt −∆φ = 0 and ψ ≤ 1, we obtain

ut −∆u− vp = aε(t + 1)a−1φ− εp(t + 1)bpφp

= [a(t + 1)a−bp−1+n(p−1)/2 − εp−1ψp−1]ε(t + 1)bp−pn/2ψ ≥ 0

for t ≥ 0, whenever

a− bp ≥ 1− n(p− 1)/2 and εp−1 ≤ a. (32.24)

Symmetrically, we have vt −∆v − uq ≥ 0 whenever

b− aq ≥ 1− n(q − 1)/2 and εq−1 ≤ b. (32.25)

Choosing a = n2 −

p+1pq−1 and b = n

2 −q+1pq−1 , the first conditions in (32.24) and

(32.25) are satisfied (with equalities) and since a, b > 0 due to max(α, β) = α < n,it suffices to choose ε > 0 small. It then follows from the comparison principle that(u, v) is global if u0 ≤ u(·, 0) and v0 ≤ v(·, 0).

32.4. Blow-up asymptotics

As compared with the scalar model problem (15.1), less is known concerning theasymptotic behavior of blowing-up solutions to system (32.1). We shall establishthe following theorem concerning type I blow-up rate for monotone solutions intime. A few results concerning other aspects of the blow-up behavior will be men-tioned in Remarks 32.12.

Theorem 32.9. Consider problem (32.1) with Ω bounded, p, q ≥ 1, pq > 1, and0 ≤ u0, v0 ∈ L∞(Ω). Assume that u, v are nondecreasing in time and (u, v) isnonstationary. Then T := Tmax(u0, v0) < ∞ and we have

C1(T − t)−α/2 ≤ ‖u(t)‖∞ ≤ C2(T − t)−α/2,

C3(T − t)−β/2 ≤ ‖v(t)‖∞ ≤ C4(T − t)−β/2,

(32.26)

284 III. Systems

in (0, T ) for some C1, C2, C3, C4 > 0.

This result was proved in [158] and its proof is based on a modification of themaximum principle arguments of [219]. The assumption ut, vt ≥ 0 is guaranteedif, for instance, 0 ≤ u0, v0 ∈ C0 ∩ C2(Ω) and ∆u0 + vp

0 ≥ 0, ∆v0 + uq0 ≥ 0 (see

Remark 52.23(ii)).

For the proof, we need the following lemmas concerning the systems of differ-ential inequalities:

y′(t) ≥ εzp,

z′(t) ≥ εyq,

(32.27)

andy′(t) ≤ zp,

z′(t) ≤ yq.

(32.28)

Lemma 32.10. Let p, q, ε > 0 satisfy pq > 1, and let 0 < T ≤ ∞. Assume that0 ≤ y, z ∈ C1(0, T ), (y, z) ≡ (0, 0), and that (y, z) solves (32.27) on (0, T ). ThenT < ∞ and there holds

y(t) ≤ C1(T − t)−α/2, z(t) ≤ C1(T − t)−β/2, 0 < t < T, (32.29)

with C1 = C1(p, q, ε) > 0.

Lemma 32.11. Let p, q > 0 satisfy pq > 1, and let 0 < T < ∞. Assume that0 ≤ y, z are locally absolutely continuous and nondecreasing on (0, T ), and that(y, z) solves (32.28) a.e. on (0, T ). Assume also that sup(0,T )(y + z) = ∞ and that(32.29) is satisfied for some C1 > 0. Then there holds

y(t) ≥ η(T − t)−α/2, z(t) ≥ η(T − t)−β/2, T − η < t < T,

with η = η(p, q, C1) > 0.

Proof of Lemma 32.10. We have

ε−p−1y(t) ≥ ε−p

∫ t

0

zp(s) ds ≥∫ t

0

(∫ s

0

yq(σ) dσ)p

ds =: h(t).

Therefore,

[(h′)(p+1)/p]′ = (p + 1)(∫ t

0

yq(s) ds)p

yq(t) ≥ (p + 1)εq(p+1)h′hq = C(hq+1)′.

Since h(0) = h′(0) = 0, it follows that (h′)(p+1)/p ≥ Chq+1. Moreover, due to(y, z) ≡ (0, 0), we may assume h > 0 on (t0, T ) for some t0 ∈ (0, T ). Puttingγ = p q+1

p+1 > 1, we get [h1−γ ]′ = −(γ − 1)h′h−γ ≤ −C < 0. Integrating over (t, s)

with t0 < t < s < T , we obtain h1−γ(t) ≥ h1−γ(s)+C(s− t) ≥ C(s− t). It followsthat T < ∞. By letting s → T , we obtain

h(t) ≤ C(T − t)−1/(γ−1) = C(T − t)−α/2, t0 < t < T. (32.30)

Next, fix t0 < t < T and let τ = (T − t)/4. Since y′ ≥ 0, we have

h(t + 2τ) =∫ t+2τ

0

(∫ s

0

yq(σ) dσ)p

ds

≥ τ(∫ t+τ

0

yq(σ) dσ)p

≥ τ(τyq(t))p = τp+1ypq(t).

In view of (32.30), we deduce

ypq(t) ≤ τ−(p+1)h(t + 2τ) ≤ Cτ−(p+1)(T − t− 2τ)−(p+1)/(pq−1)

= C(T − t)−pq(p+1)/(pq−1),

hence the estimate of y on (t0, T ). The estimate of z follows symmetrically. Sincethe constant C is independent of t0 and y = z = 0 in (0, t) if h(t) = 0, the estimatesabove (obtained in (t0, T )) remain true in (0, T ).

Proof of Lemma 32.11. We first observe that for suitable a, b > 0 (dependingon p, q) the functions

y(t) := a(T − t)−α/2, z(t) := b(T − t)−β/2

satisfy y′(t) = zp(t), z′(t) = yq(t) on (0, T ). We deduce that, for each t ∈ (0, T ),

either y(t) ≥ y(t) or z(t) ≥ z(t). (32.31)

(Indeed, if this failed for some t ∈ (0, T ), then we would have y(t) < y(t − η)and z(t) < z(t − η) for some η > 0 so that, by a simple comparison argument,y(s) ≤ y(s − η) and z(s) ≤ z(s− η), t ≤ s < T , contradicting the fact that (y, z)is unbounded on (0, T ).)

Assume for contradiction that there exist sequences ηi → 0+ and ti → T suchthat

z(ti) ≤ ηi(T − ti)−β/2.

Fix K > 1 and put t′i := ti−K(T −ti). Then (32.31), (32.29) and z′ ≥ 0 guaranteethat, for large i,

a(T−ti)−α/2 ≤ y(ti) ≤ y(t′i)+∫ ti

t′i

zp(s) ds ≤ C1(T−t′i)−α/2+Kηp

i (T−ti)1−p(β/2).

286 III. Systems

Using 1 − p(β/2) = −α/2 and noting that T − t′i = (1 + K)(T − ti), we geta ≤ C1(1 + K)−α/2 + Kηp

i . Letting i → ∞, we get a contradiction for K =K(p, q, a) large enough. Consequently, there exists η = η(p, q) > 0 such thatz(t) ≥ η(T − t)−β/2 on [T − η, T ). The estimate for y follows symmetrically.

Proof of Theorem 32.9. We first prove the upper estimates. Using the maxi-mum principle in a similar way as in the proof of Theorem 23.5, we obtain ut, vt > 0in QT and ∂νut, ∂νvt < 0 on ST . Choosing τ ∈ (0, T ) we can find ε > 0 such thatut(x, τ) ≥ εvp(x, τ) and vt(x, τ) ≥ εuq(x, τ) for all x ∈ Ω. Set f = f(v) := vp,g = g(u) := uq and J := ut − εf , H := vt − εg. Then

Jt −∆J = f ′vt − εf ′g + εf ′′|∇v|2,

henceJt −∆J ≥ f ′H in Qτ (32.32)

where Qτ := Ω× (τ, T ), and symmetrically

Ht −∆H ≥ g′J in Qτ . (32.33)

Since f ′(v) and g′(u) and nonnegative and locally bounded in Ω× [τ, T ), we mayapply the maximum principle (Proposition 52.21) to system (32.32)–(32.33). AsJ, H ≥ 0 on the parabolic boundary of Qτ , we thus have J, H ≥ 0 in Qτ . Con-sequently, ut ≥ εvp, vt ≥ εuq in Qτ . Applying Lemma 32.10 with y(t) = u(x, t),z(t) = v(x, t) (for each fixed x ∈ Ω) yields T < ∞ and the upper estimates in(32.26).

Let us turn to the lower estimates. We now set

y(t) = maxx∈Ω

u(x, t), z(t) = maxx∈Ω

v(x, t).

Arguing as in the (alternative) proof of Proposition 23.1, we obtain y′ ≤ zp andz′ ≤ yq a.e. in (0, T ). Consequently, the lower estimates in (32.26) are guaranteedby Lemma 32.11.

Remarks 32.12. (i) Blow-up rate. The blow-up estimates in Theorem 32.9were obtained before in [110] under stronger restrictions on p, q and the initial data.On the other hand, when Ω = R

n, (32.26) is valid for all nonglobal nonnegativesolutions if p, q > 1 satisfy max(α, β) ≥ n [130]. This remains true for generaldomains if max(α, β) ≥ n + 1 [199]. The proofs rely on rescaling arguments andFujita-type theorems (cf. Remark 26.12). In the case Ω = R

n and max(α, β) > n,the upper estimate was proved before in [29] by different arguments based onMoser-type iteration.

(ii) Blow-up set. Concerning the blow-up set for problem (32.1), the followinghas been proved recently in [493]. Let Ω = BR and p, q > 1. Assume that u, v ≥ 0

33. The role of diffusion in blow-up 287

are radially symmetric and decreasing in |x|. If (u, v) blows up in finite time andsatisfies the upper blow-up estimates in (32.26), then single-point blow-up occursat x = 0. If, moreover, ut, vt ≥ 0, then the final blow-up profiles satisfy the lowerestimates

u(x, T ) ≥ c1|x|−α, v(x, T ) ≥ c2|x|−β

for |x| small. In the special case p = q > 1 and n = 1, an earlier result on single-point blow-up appeared in [216]. On the other hand, fine asymptotic propertiesof blow-up solutions of (32.1), including a classification of blow-up profiles, havebeen obtained in [29], [540] when Ω = R

n under the assumption that |p− q| 1.(iii) Nonsimultaneous blow-up. For system (32.1), it is easy to see that blow-

up is always simultaneous: If T = Tmax < ∞, then both components blow up inthe sense that

lim supt→T

‖u(t)‖∞ = lim supt→T

‖v(t)‖∞ = ∞.

Indeed if u, say, were uniformly bounded on QT , then the second equation wouldyield a uniform bound on v, hence contradicting (32.3). For different systems witha weaker coupling, nonsimultaneous blow-up may occur. For instance, if thenonlinearities in (32.1) are replaced with f(u, v) = urvp, g(u, v) = vsuq, or withf(u, v) = ur + vp, g(u, v) = vs + uq, then for suitable p, q, r, s > 0 and suitablenonnegative initial data, one component may blow up while the other remainsbounded (see [466, pp. 467-472], [432], [494], [458]).

33. The role of diffusion in blow-up

In this section, we discuss the different possible effects of adding linear diffusion(and some boundary conditions) to a system of ODE’s. It will turn out that quiteopposite effects can be observed:

a. In the case of an ODE system whose solutions all exist globally, it can eitherhappen that:• diffusion preserves global existence (for all initial data),or that:• diffusion induces blow-up (for some initial data).

b. Consider the case of ODE systems for which (at least some) solutions blowup in finite time. We already know examples where the addition of diffusion (evenwith homogeneous Dirichlet conditions) will not prevent the blow-up of (some)solutions (cf. Theorem 32.5). Of course, we have encountered in Section 17 a similarsituation in the scalar case. We will see that at the opposite, for certain such ODEsystems, the addition of diffusion and homogeneous Dirichlet conditions can makeall solutions global and bounded (again, a similar example in the scalar case wasgiven in Section 19).

288 III. Systems

All the systems appearing in this section are locally well-posed under the as-sumptions that will be made on the data (this will be a consequence of Exam-ple 51.12). The existence time of the unique, maximal, classical solution is denotedby Tmax or Tmax(u0, v0), and the continuation and regularity properties (32.3) and(32.4) are valid. Also, we only consider nonnegative initial data and solutions. Onthe other hand, the systems in this section do not satisfy the comparison principlein general, and we shall need to rely on other techniques.

33.1. Diffusion preserving global existence

Let us consider the following system

ut − a∆u = f(u, v), x ∈ Ω, t > 0,

vt − b∆v = g(u, v), x ∈ Ω, t > 0,

uν = vν = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

v(x, 0) = v0(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(33.1)

where a, b are positive constants. Here Ω is either a bounded domain or the whole ofR

n (in which case the boundary conditions are of course empty), and (u0, v0) ∈ X+,defined in (32.2). We assume that f, g : [0,∞)2 → R are C1-functions and thatthey satisfy

f(0, v), g(u, 0) ≥ 0, for u, v ≥ 0, (33.2)

which ensures that system (33.1) preserves positivity. (Indeed, we may extend fto R

2 by f(u, v) = f(|u|, |v|), and (33.2) then implies ut − a∆u ≥ b1(x, t)u, whereb1 = f ′

u(θu, v), θ = θ(x, t) ∈ (0, 1), and similarly for v.)In this subsection, we consider two classes of systems of the form (33.1): systems

with dissipation of mass and systems of Gierer-Meinhardt type.

Systems with dissipation of mass

This class corresponds to nonlinearities satisfying the structure condition

f(u, v) + g(u, v) ≤ 0, for all u, v ≥ 0. (33.3)

In case Ω is bounded, condition (33.3) guarantees that system (33.1) possesses theso-called mass-dissipation property:

t →M(t) is nonincreasing, where M(t) :=∫

Ω

u(x, t) dx +∫

Ω

v(x, t) dx.

Indeed, this follows immediately by integrating the differential equations in (33.1)over Ω and using the boundary conditions. This property is natural in the context


of chemical or biological applications, where systems of this form arise. If one looksat the corresponding kinetic system without diffusion, i.e. the ODE counterpartof (33.1):

U ′ = f(U, V ), t > 0,

V ′ = g(U, V ), t > 0,

U(0) = U0 ≥ 0,

V (0) = V0 ≥ 0,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(33.4)

then it is clear that solutions of (33.4) are global since 0 ≤ U(t)+V (t) ≤ U0 + V0.A central issue is to determine whether or not the mass-dissipation structure con-dition still guarantees the global existence of solutions for the diffusive system(33.1). In the case of equal diffusions a = b, it is easy to see that the answer is yes.Indeed, the function w = u + v then satisfies

wt − a∆w = f + g ≤ 0,

so 0 ≤ u + v ≤ ‖u0‖∞ + ‖v0‖∞ by the maximum principle and global existencefollows. In the case of different diffusions a = b, a case often encountered in applica-tions, this has long remained open and has motivated a large amount of work, alongwith related questions (see e.g. the survey article [348] and references therein). Itwill turn out that the answer is no in general (see Theorem 33.12 and the precedingcomments). However, we shall now see that global existence is ensured if we makesome additional assumption.

An important particular case is that when f ≤ 0, which means that the first sub-stance is absorbed by the reaction (systems with so-called “triangular” structure).Then one immediately obtains a uniform bound for u, since

u(x, t) ≤ ‖u0‖∞, x ∈ Ω, t ∈ (0, Tmax) (33.5)

by the maximum principle. The problem is then reduced to obtaining a uniformestimate of v.

A simple case when this can be done is when a > b. This means that theabsorbed substance diffuses faster than the other one. The following result forΩ = R

n was proved in [348]. A similar result was obtained in [295] for Ω bounded,but the proof is more delicate.

Theorem 33.1. Let Ω = Rn, a > b > 0 and assume

f(u, v) ≤ 0 ≤ g(u, v), for all u, v ≥ 0, (33.6)

along with (33.2), (33.3). Then, for all (u0, v0) ∈ X+, the solution of problem(33.1) is global. Moreover, u, v are uniformly bounded in R

n × [0,∞).

The proof is based on a simple comparison property concerning the kernelsassociated with the operators ∂t − a∆ and ∂t − b∆.

290 III. Systems

Proof. Let us denote by Sλ(t) the semigroup (say, on L∞(Rn)) corresponding tothe operator ∂t − λ∆. We observe that for all 0 ≤ φ ∈ L∞(Rn),

λ → λn/2[Sλ(t)φ](x) is nondecreasing for all (x, t). (33.7)

This follows from the fact that

[Sλ(t)φ](x) = (4πλt)−n/2

∫Rn

exp[−|y|2/4λt]φ(x − y) dy.

Denoting

za(t) = −∫ t

0

Sa(t− s)f(u(s), v(s)) ds, zb(t) =∫ t

0

Sb(t− s)g(u(s), v(s)) ds,

we haveu(t) + za(t) = Sa(t)u0

hence za(t) ≤ Sa(t)u0 ≤ ‖u0‖∞. Due to f + g ≤ 0, f ≤ 0 and (33.7), it followsthat

zb(t) ≤ −∫ t

0

Sb(t− s)f(u(s), v(s)) ds ≤ (a/b)n/2za(t) ≤ (a/b)n/2‖u0‖∞.

Thereforev(t) = Sb(t)v0 + zb(t) ≤ ‖v0‖∞ + (a/b)n/2‖u0‖∞.

This along with (33.5) yields the conclusion.

Still in the case f ≤ 0 but without assuming a > b, the answer is again positiveunder a polynomial growth assumption on g:

g(u, v) ≤ C(1 + u + v)γ , for all u, v ≥ 0 and some γ ≥ 1. (33.8)

Theorem 33.2. Assume Ω bounded and let a, b > 0, a = b, and γ ≥ 1. As-sume (33.2), (33.3), (33.6) and (33.8). Then, for all (u0, v0) ∈ X+, the solution ofproblem (33.1) is global.

This result was proved in [281]. It can be shown in addition that u, v are uni-formly bounded in Ω× [0,∞).

The main ingredient of the proof is the following lemma, which guarantees thatwhenever f + g ≤ 0, v is controlled by u in Lp for any finite p. The proof is basedon a duality argument.

Lemma 33.3. Assume Ω bounded, 1 0 and T > 0. There existsC = C(T, p, a, b,Ω) > 0 such that, if u, v ∈ C2,1(Ω× (0, T ]) ∩C(QT ) satisfy

(u + v)t − a∆u− b∆v ≤ 0, x ∈ Ω, 0 < t < T,

uν = vν = 0, x ∈ ∂Ω, 0 < t < T,

(33.9)

then there holds

‖v+‖Lp(QT ) ≤ C(‖u(·, 0) + v(·, 0)‖Lp(Ω) + ‖u‖Lp(QT )

). (33.10)

Proof. Let q = p′. Fix χ ∈ D(QT ), χ ≥ 0, and let ϕ be the unique solution of theproblem

−ϕt − b∆ϕ = χ, x ∈ Ω, 0 < t < T,

ϕν = 0, x ∈ ∂Ω, 0 < t < T,

ϕ(x, T ) = 0, x ∈ Ω.

⎫⎪⎬⎪⎭ (33.11)

We have ϕ ≥ 0 by the maximum principle. Parabolic Lq-estimates (see Re-mark 48.3(ii)) guarantee

‖ϕt‖Lq(QT ) + ‖D2ϕ‖Lq(QT ) ≤ C‖χ‖Lq(QT ). (33.12)

Since ϕ(·, T ) = 0, this implies in particular

‖ϕ(·, 0)‖Lq(Ω) ≤ C‖χ‖Lq(QT ). (33.13)

Multiplying the inequality in (33.9) by ϕ, integrating by parts, and using theboundary conditions and ϕ(x, T ) = 0, we obtain

0 ≥∫ ∫

QT

(ut + vt − a∆u − b∆v)ϕdxdt

=∫ ∫

QT

(u(−ϕt − a∆ϕ) + v(−ϕt − b∆ϕ)

)dx dt

−∫

Ω

(u(x, 0) + v(x, 0))ϕ(x, 0) dx.

Therefore, by (33.12) and (33.13), we get∫ ∫QT

vχ dx =∫ ∫

QT

v(−ϕt − b∆ϕ)

≤∫ ∫

QT

u(ϕt + a∆ϕ) dx dt +∫

Ω

(u(x, 0) + v(x, 0))ϕ(x, 0) dx

≤ C(‖u‖Lp(QT ) + ‖u(·, 0) + v(·, 0)‖Lp(Ω)

)‖χ‖Lq(QT ).

292 III. Systems

Since χ ≥ 0 is arbitrary in D(QT ), the lemma follows.

Proof of Theorem 33.2. Fix r > (n+2)/2. By (33.5) and Lemma 33.3, we have

‖g(u, v)‖rLr(QT ) ≤ C

∫ T

0

∫Ω

(1 + u + v)rγ dx dt ≤ C(T ),

for all finite T ≤ Tmax. Using the variation-of-constants formula, it follows that

‖v(t)‖∞ ≤ Ct−n/2‖v0‖1 + C

∫ t

0

(t− s)−n/2r‖g(u(s), v(s))‖Lr(Ω) ds

≤ Ct−n/2‖v0‖1 + C(∫ t

0

(t− s)−n/2(r−1) ds)(r−1)/r

‖g(u, v)‖Lr(Qt)

≤ Ct−n/2‖v0‖1 + C(T )tθ,

for all 0 < t < T , with θ = 1 − (n + 2)/2r > 0. This along with (33.5) yieldsTmax = ∞.

Remark 33.4. The duality argument in the proof of Lemma 33.3 has other ap-plications. For instance, under the assumptions of Theorem 33.2, it yields globalexistence for the system with inhomogeneous Neumann boundary conditions:

ut − a∆u = f(u, v), x ∈ Ω, t > 0,

vt − b∆v = g(u, v), x ∈ Ω, t > 0,

uν = α1(t), x ∈ ∂Ω, t > 0,

vν = α2(t), x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

v(x, 0) = v0(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(33.14)

where αi are arbitrary smooth functions. This works also in the case of Dirichletboundary conditions u = α1(t), v = α2(t) [349]. The argument can also be usedto study the case of nonlinearities of the form g = −f = c(x)upvq, with sign-changing c(x) (see [282], [417]). Another system of physical interest, the so-calledBrusselator, corresponding to the choices f = −uv2 +Bv, g = uv2− (B +1)v +A,can also be handled by similar techniques [281].

If f, g do not have a sign, it is still possible to show global existence modulo theadditional dissipation condition:

λf(u, v) + g(u, v) ≤ 0, for u, v ≥ 0, (33.15)

with sufficiently large λ > 1, assuming also that f, g have at most polynomial(upper) growth:

f(u, v), g(u, v) ≤ C(1 + u + v)γ , for all u, v ≥ 0 and some γ ≥ 1. (33.16)

Theorem 33.5. Assume Ω bounded and let a, b > 0, a = b, and γ ≥ 1. Assume(33.2), (33.3), (33.15), (33.16), with

λ ≥ λ0(a, b, n, γ) :=[(a + b)2

4ab

]m−1

≥ 1, (33.17)

where m in the smallest integer such that m > n(γ− 1)/2. Then, for all (u0, v0) ∈X+, the solution of problem (33.1) is global. Moreover, u, v are uniformly boundedin Ω× [0,∞).

Theorem 33.5 is from [309]. A typical example (without sign condition) to whichit applies is given by f = upvq − urvs, g = urvs − λupvq for any p, q, r, s ≥ 1 andλ > 1 large enough (depending on p, q, r, s, n, a, b). Interestingly, it will turn outthat the largeness assumption on λ is in some sense necessary (see Theorem 33.12and the preceding paragraphs).

Remarks 33.6. (i) The proof of Theorem 33.5 is based on a suitable Lyapunovfunctional, cf. Lemma 33.7 below, whereas Theorem 33.2 was based on a dualityargument. Note that (33.15) is satisfied in particular if f + g ≤ 0 and f ≤ 0.Therefore, Theorem 33.5 is more general than Theorem 33.2. However, the dual-ity argument has other applications (cf. Remark 33.4) which do not seem to betractable by the Lyapunov functional approach. Also, in the case of homogeneousDirichlet boundary conditions, results similar to Theorem 33.5 have been obtainedin [468] by duality techniques, but the largeness condition on λ is not explicit.

(ii) Note that if γ < (n + 2)/n, then λ0 = 1 in (33.17) (with m = 1), so thatcondition (33.15) disappears. For earlier results related to Theorems 33.2 and 33.5,see [11], [12] (based on Moser’s iteration), [460] (based on bootstrap) and [350](based on a Lyapunov functional). On the other hand, the global existence resultof Theorem 33.2 has been extended to f, g satisfying some exponential growthconditions, for instance for g = −f = uev. For this we refer to [270], [52] (relyingon suitable Lyapunov functionals) and to [274], [294] (based on a delicate analysisusing parabolic BMO estimates).

The key of the proof of Theorem 33.5 is the following Lyapunov functional.

Lemma 33.7. Assume Ω bounded, a, b > 0,

K ≥ a + b

2√

ab(33.18)

and let m be any positive integer. Assume that f, g satisfy (33.2), (33.3), (33.15),(33.16), for some λ ≥ K2(m−1). Let (u, v) be a solution of (33.1) and let

L(t) =∫

Ω

Hm (u(x, t), v(x, t)) dx,

294 III. Systems

where

Hm(u, v) =m∑

i=0

CimKi2−iuivm−i, Ci

m =m!

i!(m− i)!.

Then L′(t) ≤ 0 on the interval (0, Tmax).

Proof. Set w = Kv and L1(t) = KmL(t). We have

KmHm(u, v) =m∑

i=0

CimKi2uiwm−i

and w solveswt − b∆w = Kg(u, v).

Differentiating L1 with respect to t yields

L′1(t) =

∫Ω

[ m∑i=1

(iCi

mKi2ui−1wm−i)ut +

m−1∑i=0

((m− i)Ci

mKi2uiwm−i−1)

wt

]dx

=∫

Ω

m∑i=1

(iCi

mKi2ui−1wm−i)

(a∆u + f(u, v)) dx

+∫

Ω

m∑i=1

((m− i + 1)Ci−1

m K(i−1)2ui−1wm−i)

(b∆w + Kg(u, v)) dx

=∫

Ω

[ m∑i=1

aiCimKi2ui−1wm−i∆u + b(m− i + 1)Ci−1

m K(i−1)2ui−1wm−i∆w]dx

+∫

Ω

[ m∑i=1

iCimKi2ui−1wm−if(u, v)

+ (m− i + 1)Ci−1m K(i−1)2+1ui−1wm−ig(u, v)

]dx

=: I + J.

By using Green’s formula we obtain

I = −∫

Ω

(A |∇u|2 + B∇u∇w + C |∇w|2

)dx,

where

A =m∑

i=2

ai(i− 1)CimKi2ui−2wm−i,

B =m−1∑i=1

ai(m−i)CimKi2ui−1wm−i−1+

m∑i=2

b(i−1)(m−i+1)Ci−1m K(i−1)2ui−2wm−i,


and

C =m−1∑i=1

b(m− i)(m− i + 1)Ci−1m K(i−1)2ui−1wm−i−1.

Using the fact that

iCim = mCi−1

m−1, i = 1, . . . , m,

(m− i)Cim = mCi

m−1, i = 0, . . . , m− 1,(33.19)

we get

A = am(m− 1)m∑

i=2

Ci−2m−2K

i2ui−2wm−i,

B = am(m− 1)m−1∑i=1

Ci−1m−2K

i2ui−1wm−i−1

+ bm(m− 1)m∑

i=2

Ci−2m−2K

(i−1)2ui−2wm−i

=: B1 + B2,

and

C = bm(m− 1)m−1∑i=1

Ci−1m−2K

(i−1)2ui−1wm−i−1.

Putting j = i− 2, we have

A = am(m− 1)m−2∑j=0

Cjm−2K

(j+2)2ujwm−j−2,

B2 = bm(m− 1)m−2∑j=0

Cjm−2K

(j+1)2ujwm−j−2,

and putting j = i− 1, we get

B1 = am(m− 1)m−2∑j=0

Cjm−2K

(j+1)2ujwm−j−2,

C = bm(m− 1)m−2∑j=0

Cjm−2K

j2ujwm−j−2.

296 III. Systems

Therefore,

I =−m(m− 1)m−2∑j=0

Cjm−2

∫Ω

ujwm−j−2

×(aK(j+2)2 |∇u|2 + (a + b)K(j+1)2∇u∇w + bKj2 |∇w|2

)dx.

The quadratic forms (with respect to ∇u and ∇w) are positive since((a + b)K(j+1)2

)2 − 4abKj2K(j+2)2 = K2j2+4j+2

((a + b)2 − 4abK2

)≤ 0

for j = 0, 1, . . . , m− 2, due to (33.18). It follows that I ≤ 0.On the other hand, by (33.19), we have

J = mm∑

i=1

Ci−1m−1

∫Ω

(Ki2f(u, v) + K(i−1)2+1g(u, v)

)ui−1wm−i dx.

Since (33.3) and (33.15) imply µf + g ≤ 0 for all µ ∈ [1, λ], we obtain

Ki2f(u, v) + K(i−1)2+1g(u, v) = K(i−1)2+1(K2(i−1)f(u, v) + g(u, v)) ≤ 0

for i = 1, . . . , m, hence J ≤ 0.

Proof of Theorem 33.5. In Lemma 33.7, we take K = a+b2√

aband m as in the

statement of the theorem. Then λ0 = K2(m−1) and we deduce from Lemma 33.7that u(t) and v(t) are bounded in Lm(Ω). Since m > n(γ − 1)/2, by similararguments as in the proof of Theorem 16.4, one deduces that they are boundedin L∞(Ω). (Alternatively one could use modifications of arguments in the proof of(15.2) in Theorem 15.2.) In particular, this implies Tmax = ∞ and the theorem isproved.

Remarks 33.8. (i) Simple modifications of the proofs of Theorems 33.2 and 33.5,show that global existence (without boundedness) is still true if the conditions f+gand/or λf + g ≤ 0 are replaced by f + g and/or λf + g ≤ C(1 + u + v).

(ii) Under the assumptions of Theorem 33.5, if u, v ≥ 0 and (u, v) solves (33.1)for t ∈ (0, T ), with the boundary conditions replaced by

uν ≤ 0, vν ≤ 0,

then u, v are uniformly bounded in Ω× [0, T ). This follows from a simple modifi-cation of the proof of Lemma 33.7 and Theorem 33.5.

Systems of Gierer-Meinhardt type

We consider the system

ut − a∆u = −µ1u + up/vq + σ, x ∈ Ω, t > 0,

vt − b∆v = −µ2v + ur/vs, x ∈ Ω, t > 0,

uν = vν = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

v(x, 0) = v0(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(33.20)

where p > 1, q, r, s ≥ 0, a, b > 0, µ1, µ2, σ ≥ 0, and u0, v0 ∈ C(Ω) with u0, v0 > 0.By the maximum principle, we immediately obtain the lower bounds

u(x, t) ≥(min

Ωu0

)e−µ1t, v(x, t) ≥

(min

Ωv0

)e−µ2t, x ∈ Ω, 0 < t < Tmax.

(33.21)These bounds imply in particular that

if Tmax < ∞, then lim supt→Tmax

‖u(t)‖∞ =∞.

System (33.20) (for instance with p = r = 2, q = s = 4, n = 1) arises in abiological model of pattern formation, due to [242]. Here u and v represent theconcentrations of an activator and an inhibitor, respectively. The peaks of highconcentration of activator give the positional information for the development ofa structure, e.g. a tentacle in the polyp Hydra.

An essentially complete answer to the global existence/nonexistence questionfor system (33.20) is provided by the following theorem from [333] (see also [292],[398] for related results). Earlier partial results of global existence had been provedin [460], [352]. We note also that a large amount of literature has been devotedto the singular perturbation problem associated with the study of “spike-layers”(stationary solutions developing a finite number of concentration peaks as a → 0).We refer for this to the surveys [391], [392].

Theorem 33.9. Assume Ω bounded.(i) Assume that

p− 1r

< min( q

s + 1, 1). (33.22)

Then, for all u0, v0 ∈ C(Ω) with u0, v0 > 0, the solution (u, v) of problem (33.20)is global. If in addition µ1, µ2, σ > 0, then u, v are uniformly bounded in Ω×[0,∞).(ii) Assume that

p− 1r

> min( q

s + 1, 1),

p− 1r

= 1. (33.23)

Then there exist space-independent initial data u0, v0 > 0 such that the solution(u, v) = (u(t), v(t)) of problem (33.20) satisfies Tmax < ∞.

298 III. Systems

Remark 33.10. Diffusion preserving global existence. It follows from The-orem 33.9 that, for any p, q, r, s such that the solutions of the ODE system areall global, the addition of diffusion preserves global existence (except perhaps forp−1 = r and for the equality case in (33.22), which seem to be open). No exampleof blow-up seems to be known for system (33.20) beside the space-independentsolutions.

The proof of assertion (i) relies on multiplier arguments and on the followingconsequence of Young’s inequality.

Lemma 33.11. Assume that p, q, r, s satisfy (33.22). For all η, α, β > 0, thereexist C = C(η, α, β) > 0 and θ = θ(α) ∈ (0, 1) such that

αxp−1+α

yq+β≤ β

xr+α

ys+1+β+ C

(xα

yβ

)θ

, x ≥ 0, y ≥ η. (33.24)

Proof. Let x > 0 and y ≥ η. Inequality (33.24) is equivalent to

αxp−1

yq≤ β

xr

ys+1+ C

( yβ

xα

)1−θ

.

Write

αxp−1

yq=( xr

ys+1

)(p−1)/r

y(p−1)(s+1)/r−q = C(β

xr

ys+1

)γ

y−m,

where γ = (p − 1)/r < 1 and m = q − (p − 1)(s + 1)/r > 0. For each 0 < ε <min(m/(s + 1), 1− γ), using y ≥ η and Young’s inequality, we obtain

αxp−1

yq= C

(β

xr

ys+1

)γ+ε

y−m+(s+1)εx−rε ≤ C(β

xr

ys+1

)γ+ε( yβ

xα

)rε/α

≤ βxr

ys+1+ C

( yβ

xα

)rε/(1−γ−ε)α

,

and (33.24) follows by taking ε sufficiently small.

Proof of Theorem 33.9(i). We shall only prove global existence and uniformboundedness in the case µ1, µ2, σ > 0. Global existence in the general case canbe shown by simple modifications (using the lower bound (33.21) on finite timeintervals instead of (33.25) below).

Step 1. Lower estimates. We claim that there exists c1 > 0 such that

u, v ≥ c1, x ∈ Ω, 0 < t < Tmax. (33.25)

Since u satisfies ut− a∆u > 0 on u < σ/µ1 (along with homogeneous Neumannconditions), the maximum principle implies u ≥ δ := min(σ/µ1, minΩ u0) > 0 in

Ω × [0,∞). Then, v satisfies vt − b∆v > 0 on v < (δr/µ2)1/(s+1) and the lowerbound for v follows similarly.

Step 2. Bound for a quotient. We claim that, for all large α, β > 0, the function

φ = φα,β(t) :=∫

Ω

uα

vβdx

satisfiessup

t∈(0,Tmax)

φ(t) < ∞. (33.26)

By (33.20), we have

φ′(t) =∫

Ω

(α

uα−1ut

vβ− β

uαvt

vβ+1

)dx

= α

∫Ω

uα−1

vβ

(a∆u− µ1u + σ +

up

vq

)dx− β

∫Ω

uα

vβ+1

(b∆v − µ2v +

ur

vs

)dx.

Using Green’s formula, we deduce that

φ′(t) = (−αµ1 + βµ2)φ +∫

Ω

(α

up−1+α

vq+β− β

ur+α

vs+1+β+ ασ

uα−1

vβ

)dx

+∫

Ω

(−aα(α− 1)

uα−2

vβ|∇u|2 − bβ(β + 1)

uα

vβ+2|∇v|2

+ (a + b)αβuα−1

vβ+1∇u · ∇v

)dx.

(33.27)

The last integrand can be rewritten as

Q :=[−aα(α− 1)v2|∇u|2 − bβ(β + 1)u2|∇v|2 + (a + b)αβ(v∇u) · (u∇v)

] uα−2

vβ+2.

Consequently we have Q ≤ 0, provided we assume

αβ

(α − 1)(β + 1)≤ 4ab

(a + b)2, (33.28)

which guarantees that the discriminant (a+ b)2(αβ)2−4abαβ(α−1)(β +1) of thequadratic form Q is nonpositive. Owing to (33.25), we also have

uα−1

vβ=(uα

vβ

)(α−1)/α

v−β/α ≤ C(uα

vβ

)(α−1)/α

. (33.29)

Using (33.25), Lemma 33.11, (33.29) and Holder’s inequality, we obtain

φ′(t) ≤ (−αµ1 + βµ2)φ + C

∫Ω

(uα

vβ

)θ

dx + ασ

∫Ω

uα−1

vβdx

≤ (−αµ1 + βµ2)φ + C(φθ + φ(α−1)/α)(33.30)

300 III. Systems

for some θ ∈ (0, 1).

Now assume α ≥ 2 max(1, µ2/µ1) and β ≤ 2ab/(a + b)2 ≤ 1. Then (33.28) issatisfied and, since −αµ1 + βµ2 < 0, the function

f(Y ) := (−αµ1 + βµ2)Y + C(Y θ + Y (α−1)/α)

has a largest positive zero, say Y = K. Since, by (33.30), φ′(t) < 0 wheneverφ(t) > K, we deduce easily that supt∈(0,Tmax) φ(t) ≤ max(φ(0), K), hence (33.26).Since v is bounded below, it is clear that (33.26) remains true if we enlarge β. Theclaim is proved.

Step 3. L∞-bounds. By (33.26), we have upv−q and urv−s ∈ L∞((0, Tmax),Lm(Ω)) for all m < ∞. By a simple argument using the variation-of-constantsformula and the Lp-Lq-estimate (Proposition 48.4), we deduce that u and v areuniformly bounded and that Tmax = ∞.

Proof of Theorem 33.9(ii). We consider space-independent solutions of (33.20),i.e. solutions of the corresponding ODE system without diffusion. For spatiallyhomogeneous initial data u0, v0 ≥ 1 to be determined later, we assume for contra-diction that Tmax(u0, v0) > 1. In what follows, all the positive constants C, c, . . .are independent of u0, v0.

For fixed α, β > 0, let λ = αµ1 − βµ2 and w(t) = uα/vβ. By direct calculationusing (33.20) (cf. the first line of (33.27)), we have

w′ + λw = αup−1+α

vq+β− β

ur+α

vs+1+β+ ασ

uα−1

vβ. (33.31)

We consider two cases separately.Case 1: p − 1 > r. We apply (33.31) with α = 1. Taking β large enough and

using (33.21) and v0 ≥ 1, we have for all t ∈ [0, 1],

α

2up

vq+β=

α

2

( ur+1

vs+1+β

)p/(r+1)

vk ≥ βur+1

vs+1+β− C,

where k = (s + 1 + β) pr+1 − q − β > 0, and

α

2up

vq+β=

α

2

( u

vβ

)p

vm ≥ c( u

vβ

)p

,

where m = (p− 1)β − q > 0. It follows that

w′ ≥ cwp − λw − C.

Taking w(0) large enough, this implies blow-up of u before t = 1; a contradiction.

Case 2: p− 1 < r, (p− 1)(s + 1) > qr. We claim that that there exist constantsC1, C2 > 0 such that, if

ur−p+10 ≥ C1v

s+1−q0 , (33.32)

thenur−p+1 ≥ C2v

s+1−q, 0 < t ≤ 1. (33.33)

To prove this, letting z = eλtw and applying (33.31) with

α = r − p + 1 > 0 and β = s + 1− q > 0,

we see that, for all t ∈ [0, 1],

z′(t) ≤ 0 =⇒ ur+α

vs+1+β≥ up−1+α

vq+βαβ−1 =⇒ z(t) ≥ e−|λ| u

α

vβ≥ e−|λ|αβ−1 =: C1.

Consequently we have z(t) ≥ min(C1, w(0)) on [0, 1], and the claim follows withC2 = e−|λ|C1.

Now assume (33.32). Using the first equation in (33.20) and (33.33), we deducethat

u′ + µ1u ≥up

vq≥ cup−q(r−p+1)/(s+1−q) = cuγ , 0 < t ≤ 1,

where γ = 1 + (p−1)(s+1)−qrs+1−q > 1. But, taking u0 larger, this implies blow-up of u

before t = 1; a contradiction.

33.2. Diffusion inducing blow-up

In this subsection we show that certain parabolic systems admit a blowing-upsolution for some particular initial data, although the corresponding system ofODE’s has only global bounded solutions. We shall give three different examples,each of them involving a different method.

We first consider systems of the form

ut − a∆u = f(u, v), x ∈ Ω, t > 0,

vt − b∆v = g(u, v), x ∈ Ω, t > 0,

uν = α1(t), x ∈ ∂Ω, t > 0,

vν = α2(t), x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

v(x, 0) = v0(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(33.34)

under the mass-dissipation structure condition f + g ≤ 0. Sufficient conditionsensuring global existence for such systems were studied in the previous subsection.

302 III. Systems

Recall that, under a polynomial growth assumption on the nonlinearities, globalexistence of nonnegative solutions is true if f + g ≤ 0 and if either:

• λf + g ≤ 0 with λ > 1 large enough, under homogeneous boundary conditions(or more generally uν, vν ≤ 0); or

• f ≤ 0, with arbitrary (smooth) functions αi (or also under Dirichlet-Dirichletboundary conditions).

The following result [417] shows that in case of unequal diffusions, the conditionf + g ≤ 0 is not sufficient to ensure global existence, even if the additional dissi-pation property λf + g ≤ 0 is also satisfied (with some λ > 1 not too large) andf and g have polynomial growth. We point out that the example below involvesnonnegative solutions and functions αi ≤ 0, so that the condition f + g ≤ 0 stillguarantees the mass-dissipation property (d/dt)

∫Ω(u(t) + v(t)) dx ≤ 0. The result

has to be compared with Theorem 33.5, which is therefore in a sense optimal.

Theorem 33.12. Let Ω = B1 ⊂ Rn. There exist constants a, b, T > 0, a = b,

functions f, g ∈ C∞(R2, R) and α1, α2 ∈ C∞([0, T ], R), satisfying α1, α2 ≤ 0,

f + g ≤ 0, λf + g ≤ 0, for all u, v ≥ 0 and some λ > 1, (33.35)

f(u, v), g(u, v) ≤ C(1 + u + v)γ , for all u, v ≥ 0 and some γ ≥ 1, (33.36)

and such that for some C∞ initial data u0, v0 ≥ 0, system (33.34) admits a classicalnonnegative solution (u, v) on (0, T ), with

limt→T

u(0, t) = limt→T

v(0, t) = ∞.

Moreover, u and v blow up only at x = 0 as t→ T .

The proof is based on the construction of an explicit solution, of self-similarform, and involves some relatively heavy numerical computations (still doable byhand, but the construction was initially carried out with the help of the formalcomputation software Maple).

Sketch of proof (for n = 10). An explicit solution is searched under the form

u(x, t) =A(T − t) + B|x|2(T − t + |x|2)5/4

, v(x, t) =C(T − t) + D|x|2(T − t + |x|2)5/4

,

with constants A, B, C, D > 0 to be determined. Note that this is actually a self-similar solution, since it can be rewritten under the form

u = (T − t)−1/4U(y), v = (T − t)−1/4V (y), y = x(T − t)−1/2,

with

U(y) =A + B|y|2

(1 + |y|2)5/4, V (y) =

C + D|y|2(1 + |y|2)5/4

.

A direct calculation yields

ut − a∆u = (T − t)−5/4 A1 + B1|y|2 + C1|y|4(1 + |y|2)(5/4)+2

and

vt − b∆v = (T − t)−5/4 A2 + B2|y|2 + C2|y|4(1 + |y|2)(5/4)+2

,

where A1, B1, C1 and A2, B2, C2 are computed in terms of n, a, A, B and n, b, C, D,respectively. As for the functions f, g, one looks for polynomials, homogeneous andof total degree 5, of the form

f(u, v) =5∑

i=0

λiu5−ivi, g(u, v) =

5∑i=0

µiu5−ivi.

The PDE’s in system (33.34) then become equivalent to

5∑i=0

λi(A + B|y|2)5−i(C + D|y|2)i = (1 + |y|2)3(A1 + B1|y|2 + C1|y|4) (33.37)

and

5∑i=0

µi(A + B|y|2)5−i(C + D|y|2)i = (1 + |y|2)3(A2 + B2|y|2 + C2|y|4). (33.38)

Choosing n = 10 (other choices are possible) and a = 1, b = 1/10, A = 1/25,B = 1, C = 11/2, D = 1/10, it turns out that it is possible to adjust the constantsλi, µi in such a way that (33.37), (33.38) be satisfied, with moreover λi + µi < 0,so that

λf + g =5∑

i=0

(λλi + µi)u5−ivi ≤ 0

for λ equal or close to 1. Finally, for r = |x| = 1, we compute

α1(t) = uν(x, t) = ur(1, t) =(4B − 5A)(T − t)−B

2(T − t + 1)9/4

and an analogous expression for α2(t) (with C, D in place of A, B). Taking T > 0small enough, it follows that αi(t) ≤ 0 on [0, T ].

304 III. Systems

Remark 33.13. (i) Other examples of blow-up. An example similar to thatof Theorem 33.12 is also constructed in [416] for nonlinearities f(x, t, u, v) =c1(x, t)upvq, g(x, t, u, v) = c2(x, t)upvq, with n = 1, p, q > 1 and sign-changingfunctions ci such that c1+c2 ≤ 0. However, it remains an open problem to constructsimilar examples of blow-up in the case of homogeneous boundary conditions. Onthe other hand, beyond the special examples, there is a lack of general blow-upcriteria for such systems, as well as of a description of possible singularities, incomparison with the scalar problems studied in Chapter II.

(ii) Global weak solutions. Consider problem (33.1) under the assumptions(33.2), (u0, v0) ∈ X+, f + g ≤ 0 and λf + g ≤ 0 for some λ > 1. Theorem 33.12suggests that this problem need not admit a global classical solution. However, itwas shown in [415] that there exists a global weak solution in some appropriate L1-sense. Also, it is to be noted that in the example constructed in Theorem 33.12,it is possible to extend the solution across the blow-up time to a global weaksolution.

Still for mass-dissipative systems of the form (33.34), here with f ≤ 0 ≤ g andf +g = 0, the next result [64], [65] shows that mixed Dirichlet-Neumann conditionscan lead to finite-time blow-up, even for equal diffusions. Namely we consider theone-dimensional problem

ut − uxx = −uvp, x ∈ (−1, 1), t > 0,

vt − vxx = uvp, x ∈ (−1, 1), t > 0,

u(±1, t) = 1, t > 0,

vx(±1, t) = 0, t > 0,

u(x, 0) = u0(x), x ∈ (−1, 1),

v(x, 0) = v0(x), x ∈ (−1, 1).

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(33.39)

Theorem 33.14. Assume p > 2. Let u0, v0 ∈ C2([−1, 1]) be even and satisfyu0(±1) = 1, (v0)x(±1) = 0,

0 < u0 ≤ 1, v0 > 0 in [−1, 1],

(u0 + v0)x, (v0)x ≥ 0 in [0, 1],

(u0 + v0)xx ≥ 0, (v0)xx + u0vp0 ≥ 0 in [−1, 1]. (33.40)

Then the solution of (33.39) satisfies Tmax <∞ and limt→Tmax v(1, t) = ∞.

Remark 33.15. It has been shown in [349] that solutions of (33.39) exist globallyif 0 0 (this follows from a


simple modification of the proof of Theorem 33.2 and Lemma 33.3). Analogues ofTheorem 33.14 in higher dimension can be found in [65].

The proof of Theorem 33.14 is based on monotonicity and subsolution argu-ments — to obtain pointwise lower bounds for u, v — and on the use of a simpledifferential inequality.

Proof of Theorem 33.14. Step 1. Absence of steady states. We easily verify that(33.39) has no nonnegative stationary solution except (U, V ) ≡ (1, 0). Indeed, if(U, V ) is a nonnegative stationary solution, then Vxx = −UV p ≤ 0 and Vx(±1) =0, hence Vx ≡ 0, so that V ≡ 0 (since U ≡ 0). But then Uxx ≡ 0, hence U ≡ 1 dueto U(±1) = 1.

Step 2. Monotonicity properties. From the assumptions on u0, v0, the functionsu, v are symmetric in x, and we have 0 ≤ u ≤ 1. We next observe that (w, v), withw := u + v, solves the equivalent system

wt − wxx = 0, x ∈ (−1, 1), t > 0,

vt − vxx = (w − v)vp, x ∈ (−1, 1), t > 0,

w(±1, t) = 1 + v(±1, t), t > 0,

vx(±1, t) = 0, t > 0,

w(x, 0) = (u0 + v0)(x), x ∈ (−1, 1),

v(x, 0) = v0(x), x ∈ (−1, 1).

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(33.41)

Now we claim that

wt, vt ≥ 0 in [−1, 1]× [0, Tmax) and wx, vx ≥ 0 in [0, 1]× [0, Tmax). (33.42)

Let (y, z) = (wt, vt). By continuous dependence, it suffices to prove that y, z ≥ 0when the second inequality in (33.40) is assumed to be strict. By continuity, wehave z > 0 in [−1, 1] for t small. Assume for contradiction that there is a first t0 > 0such that z(x0, t0) = 0 for some x0 ∈ [−1, 1], and denote Q0 := [0, 1]× [0, t0] andS0 := −1, 1 × [0, t0]. Since

yt − yxx = 0 in Q0,

y(·, 0) ≥ 0 and y = z ≥ 0 in S0, the maximum principle implies y ≥ 0 in Q0. Butwe then have

zt − zxx = vpy + b(x, t)z ≥ b(x, t)z in Q0,

with b = (pw − (p + 1)v)vp−1, and z ≥ 0 in Q0. Therefore, x0 = ±1 by the strongmaximum principle (since z(·, 0) ≡ 0). But this is impossible in view of Hopf’slemma, since zx = 0 on S0. We have thus proved the first part of (33.42).

306 III. Systems

Next, we have wxx = wt ≥ 0 and wx(0, t) = 0. Therefore wx ≥ 0 in [0, 1] ×[0, Tmax). By differentiating the second equation of (33.41) in x, we see that h := vx

satisfies

ht − hxx = vpwx + b(x, t)h ≥ b(x, t)h in [0, 1]× [0, Tmax),

with h(0, t) = h(1, t) = 0. The second part of (33.42) then follows from the maxi-mum principle.

We now denoteM(t) := max

[−1,1]×[0,t]v = v(1, t)

and we assume for contradiction that Tmax = ∞.Step 3. Unboundedness of v. We claim that

M(t)→∞, t→∞.

Otherwise u, v are bounded (recall that u ≤ 1) and since w, v are nondecreasingin time by Step 2, there would exist bounded functions (W, V ) such that

W (x) = limt→∞w(x, t), V (x) = lim

t→∞ v(x, t).

But the monotonicity of w, v guarantees that (W, V ) is a stationary solution of(33.41) (see Proposition 53.8). Letting U := W − V , (U, V ) is thus a stationarysolution of (33.39), hence V ≡ 0 by Step 1. This is a contradiction, since V ≥ v0 >0.

Step 4. Pointwise lower bounds for u and v and differential inequality. For fixedT > 0, put M = M(T ), δ = min[−1,1] u0 ≤ 1 and u(x, t) = δeMp/2(x−1). Then usatisfies

ut − uxx + Mpu = 0 ≤ ut − uxx + Mpu, x ∈ (−1, 1), 0 < t < T,

u(±1, t) ≤ 1, 0 < t < T,

u(x, 0) ≤ u0(x), x ∈ (−1, 1).

⎫⎪⎬⎪⎭It follows from the maximum principle that u ≤ u in [−1, 1] × [0, T ]. We deducethat, for all t large,

u(x, t) ≥ η := δ/e > 0, x0(t) ≤ x ≤ 1, (33.43)

withx0(t) := 1−M−p/2(t) ∈ (−1, 1).

On the other hand, we have vxx = vt − uvp ≥ −vp ≥ −Mp(t). Consequently, byTaylor expansion, for some ξ ∈ (x, 1), we have

v(x, t) = v(1, t) + vx(1, t)(x − 1) + vxx(ξ, t)(x − 1)2

2≥ M(t)−Mp(t)

(x− 1)2

2.


Therefore, for t large,

v(x, t) ≥ M(t)/2, x0(t) ≤ x ≤ 1. (33.44)

Integrating the second equation in (33.39) over (−1, 1) and using (33.43), (33.44),we see that φ(t) :=

∫ 1

−1v(t) dx satisfies

φ′(t) =∫ 1

−1

uvp(t) dx ≥ (1− x0(t))η(M/2)p(t) = CMp/2(t).

Since on the other hand φ(t) ≤ 2M(t), we obtain

φ′(t) ≥ Cφp/2(t)

for t large. Since p > 2, this contradicts Tmax = ∞.

In our last example, we consider a system without the structure f + g ≤ 0,but with homogeneous Neumann conditions (unlike in the previous two examples)and equal diffusions, and for which blowing-up solutions exist for some particularinitial data, although the corresponding system of ODE’s has only global boundedsolutions.

Namely, we consider the system

ut − duxx = h(u, v)(1 + u)− δu, x ∈ (−1, 1), t > 0,

vt − dvxx = −h(u, v)(1 + v)− δv, x ∈ (−1, 1), t > 0,

ux = vx = 0, x = ±1, t > 0,

u(x, 0) = u0(x), x ∈ (−1, 1),

v(x, 0) = v0(x), x ∈ (−1, 1),

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(33.45)

with d > 0 and δ ≥ 0. Here the function h : [0,∞)2 → R, of class C1, is assumedto satisfy:

h(u, v) = −h(v, u), (33.46)

h(u, 0) = h(0, v) = 0, (33.47)

h(u, v) ≥ 0, u ≥ v ≥ 0, (33.48)

h(u, v) ≥ k(u− v)γ , u ≥ v ≥ 1, (33.49)

for some k, γ > 0. These assumptions apply for instance to the function h(u, v) =(uv)m|u − v|p(u − v), for any m ≥ 1, p ≥ 0. As for the initial data (u0, v0), weassume

u0, v0 ∈ C1([−1, 1]), u0, v0 > 0 in [−1, 1], (u0)x, (v0)x = 0, x = ±1, (33.50)

308 III. Systems

v0(x) = u0(−x), (33.51)

u0 ≥ v0 in [0, 1]. (33.52)

Due to (33.47) and the maximum principle, we have u, v > 0 in [−1, 1]× [0, T ).Let us first observe that for the corresponding system of ODE’s

y′ = h(y, z)(1 + y)− δy,

z′ = −h(y, z)(1 + z)− δz,

y(0) = y0 ≥ 0, z(0) = z0 ≥ 0,

⎫⎪⎬⎪⎭all solutions are global, and that they decay exponentially to (0, 0) if moreoverδ > 0. This follows immediately from the fact that (y + z + yz)′ = y′(1 + z)+(1 + y)z′ = −δ(y + z + 2yz).

The following result essentially comes from [527], where it was given for h(u, v) =uv(u − v). We here present the simplified proof from [492] (with more generalnonlinearities).

Theorem 33.16. Assume (33.46)–(33.52). There exists C = C(d, δ, k, γ) > 0such that, if∫ 1

0

(u0 − v0) sin(πx/2) dx ≥ C and∫ 1

−1

log(1 + u0) dx ≥ C, (33.53)

then T := Tmax(u0, v0) <∞.

The idea of the proof is to derive differential inequalities for two different func-tionals on some interval (0, T0). Integrating them yields upper estimates for themeasures of two complementary subsets of (0, T0), whose sum is less than T0,leading to a contradiction with existence up to t = T0.

Remarks 33.17. Unequal diffusions. (i) The fact that the diffusion coeffi-cients are equal in the two equations is used crucially in the proof (via the signand symmetry properties of the two components). An example of a system withblow-up induced by unequal diffusions and homogeneous Neumann conditions canbe found in [383]. The proof therein is more delicate. On the other hand, it isunknown whether or not Theorem 33.16 remains true in the case of homogeneousDirichlet boundary conditions. As for the asymptotic blow-up behavior of solutionsof (33.45), this is an essentially open problem.

(ii) In the fundamental article [516], it had been shown that unequal diffusionscan destabilize an otherwise stable constant equilibrium (global existence beinghowever preserved). Other results on diffusion-induced blow-up can be found in[133], [387], [259].

Proof of Theorem 33.16. First note that, since (u, v) := (v(−x, t), u(−x, t))solves the same system due to (33.46), (33.51), we have by uniqueness:

v(x, t) = u(−x, t). (33.54)

Next, we put

m(t) = minx∈[−1,1]

u(x, t) = minx∈[−1,1]

v(x, t), M(t) = maxx∈[−1,1]

u(x, t) = maxx∈[−1,1]

v(x, t),

and we claim thatM(t) ≥ 1, 0 < t < min(T, δ−1). (33.55)

Indeed, by adding the equations for u and v, we get

(u + v)t − d(u + v)xx = h(u, v)(u − v)− δ(u + v).

Integrating and using the boundary conditions, we deduce that

d

dt

∫ 1

−1

(u + v) dx ≥ −δ

∫ 1

−1

(u + v) dx,

hence

M(t) ≥ 12

max[−1,1]

(u + v) ≥ 14

∫ 1

−1

(u + v) dx ≥ 14e−δt

∫ 1

−1

(u0 + v0) dx,

and (33.55) follows by taking C ≥ 2e in (33.53).Now we derive two differential inequalities for the auxiliary functions φ and ψ,

defined as follows:

φ(t) := e(δ+λd)t

∫ 1

−1

wϕdx, ψ(t) :=∫ 1

−1

z dx + 2δ(t− T1), 0 ≤ t < T,

wherew = u− v, z = log

(1 + u

2

),

and

λ := π2/4, ϕ(x) = (π/4) sin(πx/2), T1 = min(δ−1, (γ(δ + λd))−1

).

We also put T0 = min(T, T1) and we set

E = t ∈ (0, T0) : m(t) ≥ 1, F = (0, T0) \ E.

Claim 1. We have

φ(t) > 0 and φ′(t) ≥ ke−1 φ1+γ χE , 0 < t < T0. (33.56)

To show this, we subtract the equations for u and v to obtain

wt − dwxx = h(u, v)(2 + u + v)− δw. (33.57)

310 III. Systems

Note in particular that since w(0, t) = wx(1, t) = 0, (33.48), (33.52) and themaximum principle imply w ≥ 0 on [0, 1] × (0, T ). Therefore, φ ≥ 0 on (0, T )and h(u, v)ϕ ≥ 0 on [−1, 1]× (0, T ) by (33.54) and (33.46). Multiplying by ϕ andintegrating by parts yields

d

dt

∫ 1

−1

wϕdx = d[wxϕ−ϕxw

]1−1

+∫ 1

−1

h(u, v)(2+u+v)ϕdx− (δ+λd)∫ 1

−1

wϕdx.

Since h(u, v)(2+u+v)ϕ ≥ h(u, v)(u−v)|ϕ|, by using (33.46), (33.48), (33.49) andJensen’s inequality we get

φ′(t) ≥ e(δ+λd)t k

∫ 1

−1

|u− v|1+γ |ϕ| dx ≥ e−γ(δ+λd)t k φ1+γ ≥ e−1 k φ1+γ , t ∈ E,

and φ′(t) ≥ 0 if t ∈ E. This, along with φ(0) ≥ C > 0 (cf. (33.53)), proves theclaim.

Claim 2. We have

ψ(t) > 0 and ψ′(t) ≥ d

8ψ2 χF , 0 < t < T0. (33.58)

By a simple computation, we get

zt − dzxx = h(u, v) + d(zx)2 − δu

1 + u.

Since h(u(x, t), v(x, t)) is odd due to (33.46) and (33.54), we have∫ 1

−1 h(u, v) dx = 0hence,

ψ′(t) =d

dt

∫ 1

−1

z dx + 2δ ≥ d

∫ 1

−1

(zx)2 dx ≥ 0. (33.59)

Since ψ(0) =∫ 1

−1log(1 + u0) dx − 2 log 2− 2δT1 > 0 by taking C > 2 log 2 + 2δT1

in (33.53), it follows in particular that ψ > 0. Now, if t ∈ F , i.e. m(t) < 1,then (33.55) implies the existence of ξ(t) ∈ [−1, 1] such that u(ξ(t), t) = 1, hencez(ξ(t), t) = 0. Therefore(∫ 1

−1

|z| dx)2

≤ 4(max |z(x, t)|)2 ≤ 4(∫ 1

−1

|zx| dx)2

≤ 8∫ 1

−1

(zx)2 dx. (33.60)

Since∫ 1

−1 z dx ≥ ψ on [0, T0) by the definition of ψ, (33.58) follows from (33.59)and (33.60).

To complete the proof of Theorem 1, we integrate (33.56) and (33.58), to obtain

φ−γ(0) ≥ γ

∫ T0

0

φ′φ−1−γ ds ≥ γke−1|E|, ψ−1(0) ≥∫ T0

0

ψ′ψ−2 ds ≥ d

8|F |.

We deduce that

min(T, δ−1, [γ(δ + λd)]−1) = T0 = |E|+ |F | ≤ (γk)−1eφ−γ(0) + 8d−1ψ−1(0).

We conclude that if φ(0) and ψ(0) ≥ C(d, δ, k, γ) > 0 large enough, then T ≤(γk)−1eφ−γ(0) + 8d−1ψ−1(0) <∞.

33.3. Diffusion eliminating blow-up

In this subsection we consider the following system

ut − d1∆u = f(u− v), x ∈ Ω, t > 0,

vt − d2∆v = f(u− v)− v, x ∈ Ω, t > 0,

u = v = 0, x ∈ ∂Ω, t > 0,

⎫⎪⎬⎪⎭ (33.61)

where Ω ⊂ Rn is bounded, d1, d2 > 0, d1−d2 > 1/λ1, f(w) = |w|p−1w, 1 < p < pS ,

and the initial data belong to Z := H10×H1

0 (Ω). We also consider the correspondingsystem of ODE’s

Ut = f(U − V ),

Vt = f(U − V )− V.

(33.62)

The following theorem is due to [197].

Theorem 33.18. Let the assumptions above be satisfied. Then:(i) there exists a solution of (33.62) which blows up in finite time;(ii) for all (u0, v0) ∈ Z, the solution of (33.61) is global and converges to the trivialsolution (0, 0) in Z as t→∞.

Proof. (i) Denote W := U−V and assume V (0) > 1, W (0) >(

p+12 V (0)2

)1/(p+1).We will prove that the solution (U, V ) blows up in finite time.

Since V ′ = f(W ) − V and W ′ = V , the functions W, V remain positive. Mul-tiplying the equation W ′′ + W ′ = W p by W ′ we see that the function E(t) :=12 (W ′(t))2 − 1

p+1W (t)p+1 is nonincreasing, hence E(t) ≤ E(0) < 0. In particular,

W p+1 >p + 1

2(W ′)2 ≥ (W ′)2 = V 2,

hence V ′ = W p−V > V 2p/(p+1)−V . Since V (0) > 1, the last differential inequalityguarantees blow-up of V .

(ii) Similarly as in Example 51.27 we get that problem (33.61) is well-posed inY := Lp+1×Lp+1(Ω). In addition, if the initial data (u0, v0) ∈ Z and the solution isbounded in Y , then this solution is global and its trajectory is relatively compact inZ, see Example 51.38. Finally, ut, vt ∈ C1((0,∞), L2(Ω))∩C((0,∞), H2∩H1

0 (Ω)).Fix (u0, v0) ∈ Z and set w := u− v. Then (w, v) solves the problem

wt − d1∆w = (d1 − d2)∆v + v, x ∈ Ω, t > 0,

vt − d2∆v = −v + |w|p−1w, x ∈ Ω, t > 0,

w = v = 0, x ∈ ∂Ω, t > 0.

⎫⎪⎬⎪⎭ (33.63)

312 III. Systems

Let −A denote the Dirichlet Laplacian in L2(Ω). Due to d1 − d2 > 1/λ1,(d1 − d2)A− 1 is a positive self-adjoint operator and its inverse

K := ((d1 − d2)A− 1)−1

is compact, positive and commutes with both A and A1/2. The first equation in(33.63) can be rewritten as

v = K(d1∆w − wt).

Now the second equation in (33.63) guarantees

K(d1∆wt − wtt) = d2K∆(d1∆w − wt)−K(d1∆w − wt) + |w|p−1w. (33.64).

Define the norm

‖ϕ‖−1 := ‖K1/2ϕ‖L2(Ω) for ϕ ∈ L2(Ω).

Multiplying (33.64) by wt and integrating in x over Ω, we have

−d1‖A1/2wt‖2−1 −12

d

dt‖wt‖2−1

=d1d2

2d

dt‖Aw‖2−1 + d2‖A1/2wt‖2−1 +

d1

2d

dt‖A1/2w‖2−1

+ ‖wt‖2−1 +1

p + 1d

dt

∫Ω

|w|p+1dx,

which impliesd

dtL(t) = −(d1 + d2)‖A1/2wt‖2−1 − ‖wt‖2−1 ≤ 0, (33.65)

where

L(t) :=12‖wt‖2−1 +

d1d2

2‖Aw‖2−1 +

d1

2‖A1/2w‖2−1 +

1p + 1

∫Ω

|w|p+1dx.

Consequently, L is a Lyapunov functional (see Appendix G) and the functionw(t) stays bounded in Lp+1(Ω). Now the second equation in (33.63) and a simpleestimate based on the variation-of-constants formula shows that v(t) stays boundedin W 2−ε,(p+1)/p(Ω) for any ε > 0 and t ≥ t0 > 0. Since this space is embedded inLp+1(Ω) for ε small due to p < pS , we see that the solution (u(t), v(t)) remainsbounded in Y . Consequently, it exists globally and is relatively compact in Z.Consequently, the ω-limit set ω(u0, v0) of this solution is a compact nonemptyconnected and invariant set in Z (see Proposition 53.3). Fix (u0, v0) ∈ ω(u0, v0)and let (u, v) be the solution of problem (33.61) with initial data (u0, v0). Setw = u − v. Since the Lyapunov functional L is constant on ω(u0, v0), (33.65)guarantees wt = 0, hence wtt = 0. Now multiplying (33.64) (with w replaced byw) with w and denoting by (·, ·) the scalar product in L2(Ω) we obtain

d1d2(KAw, Aw) + d1(KA1/2w, A1/2w) +∫

Ω

|w|p+1 dx = 0,

which implies w = 0. Now the first equation in (33.63) shows v = 0, hence u = 0.This concludes the proof.

Chapter IV

Equations with Gradient Terms

34. Introduction

In Chapter IV, we consider problems with nonlinearities depending on u and itsspace derivatives:

ut −∆u = F (u,∇u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (34.1)

Here F = F (u, ξ) : R× Rn → R is a C1-function (except for problem (34.4) with

1 < q < 2, see below).In Sections 35–39, we consider perturbations of the model problem (15.1) by

terms involving first-order derivatives:

ut −∆u = up + g(u,∇u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (34.2)

We will only consider nonnegative solutions of (34.2) (but up can be interpretedas |u|p−1u for definiteness). In many results, g might depend also on x, t, but werestrict to (34.2) for simplicity. Typical examples that we shall pay a particularattention to, are given by:

ut −∆u = up − µ|∇u|q, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (34.3)

with p, q > 1, µ > 0 (dissipative gradient term) and

ut −∆u = up − a · ∇(uq), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (34.4)

with p > 1, q ≥ 1, a ∈ Rn (convective gradient term). A motivation for studying

(34.3), (34.4) is to investigate the effect of a dissipative or convective gradient term

314 IV. Equations with Gradient Terms

on global existence or nonexistence of solutions, and on their asymptotic behavior,in finite or infinite time. We refer to [484], [490] for surveys on equations of theform (34.2). It will turn out that problems of this form reveal a number of inter-esting, qualitatively new phenomena, in comparison with the unperturbed modelproblem, such as new critical exponents, or changes in the parameters involved inthe asymptotic blow-up behavior.

In Sections 40 and 41, we consider problems whose essential superlinear char-acter comes from the gradient term. A simple model case is given by:

ut −∆u = |∇u|p, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (34.5)

with p > 1. Problem (34.5) is often referred to as a viscous Hamilton-Jacobiequation. Also, (34.5) is related with the Kardar-Parisi-Zhang equation in thephysical theory of growth and roughening of surfaces (see [252], [72] for detailsand references). Note that it is one of the simplest examples of a parabolic PDEwith a nonlinearity depending on the first-order spatial derivatives, and can thusbe considered as an analogue of the model problem (15.1). The case where thenonlinearity is replaced by um|∇u|p will also be studied. We will see that theseequations exhibit phenomena qualitatively different from (15.1), such as (boundaryor interior) gradient blow-up.

35. Well-posedness and gradient bounds

Throughout Chapter IV we denote

X := u ∈ BC1(Ω) : u = 0 on ∂Ω, (35.1)

equipped with the norm

‖w‖X := ‖w‖∞ + ‖∇w‖∞,

and X+ := w ∈ X : w ≥ 0. Problem (34.1), with F of class C1, is locally well-posed in X (see Remark 51.11). As for problem (34.4), with Ω bounded or Ω = R

n

for simplicity, it is also locally well-posed in X for all q ≥ 1 (see Example 51.15and Proposition 51.16). In particular,

if Tmax = Tmax(u0) < ∞, then limt→Tmax

‖u(t)‖X =∞. (35.2)

Moreover the solution enjoys the regularity property

u ∈ BC2,1(Ω× [t1, t2]), 0 < t1 < t2 < Tmax(u0) (35.3)

35. Well-posedness and gradient bounds 315

(cf. the corresponding proof in Example 51.9). Furthermore, problem (34.1) admitsa comparison principle, cf. Propositions 52.6, 52.10 and Remarks 52.11. In the caseof problem (34.4), see Proposition 52.16. Those results will be frequently usedthroughout this chapter without explicit reference. In particular, if F (0, 0) ≥ 0and u0 ∈ X+, then we have u ≥ 0. On the other hand, in the case of problem(34.3) in a ball or in R

n, if u0 is radial (resp. radial nonincreasing), then u enjoysthe same property, as a consequence of Proposition 52.17.

In the case of problems (34.3)–(34.5) well-posedness may actually hold true insome larger spaces, but this question is not our main concern in this chapter.However, in view of the study of the large time behavior, it will be very usefulto know weaker continuation properties than (35.2). In the case of the generalproblem (34.1) this requires some structure assumptions on F . A rather sharpresult in that direction is given by the following theorem. Here, for k > 0, wewrite F ≤ O(|ξ|k) if F (u, ξ) ≤ C(u)(|ξ|k + 1) and F ≤ o(|ξ|k) if for all ε > 0,F (u, ξ) ≤ ε|ξ|k + Cε(u), where C(u) and Cε(u) remain bounded on bounded setsof u ≥ 0.

Theorem 35.1. Consider problem (34.1) with F (0, 0) ≥ 0 and F = f + g, wheref, g ∈ C1 satisfy

|f | ≤ O(|ξ|2), |fξ| ≤ O(|ξ|), |fu| ≤ o(|ξ|2) (35.4)

and

g(0, ξ) ≤ 0, gu ≤ 0, ξ · ∂

∂ξ

( g

|ξ|)≤ 0, for all u ≥ 0, ξ ∈ R

n \ 0.(35.5)

Let u0 ∈ X+. If Tmax(u0) <∞, then

limt→Tmax(u0)

‖u(t)‖∞ = ∞.

Theorem 35.1 is a consequence of the following Bernstein-type gradient estimatefrom [59], which provides a pointwise a priori estimate of ∇u assuming a boundon u.

Proposition 35.2. Let T > 0 and assume that F = f +g, where f, g ∈ C1 satisfy(35.4) and (35.5). Let u ∈ C2,1(QT ) ∩ C(QT ), with ∇u ∈ C(QT ) ∩ L∞(QT ), be asolution of (34.1), such that

0 ≤ u ≤M in QT and |∇u0| ≤ M in Ω (35.6)

for some 0 < M <∞. Then there holds

|∇u| ≤ C = C(M, T, F, Ω) in QT .

Remarks 35.3. (a) Theorem 35.1 reduces the proof of global existence to thederivation of a uniform estimate of u (on bounded time intervals). It also guaran-tees that finite-time blow-up, in case it occurs, takes place in the L∞-norm.

(b) Assumptions (35.4), (35.5) in Theorem 35.1 can be viewed as one-sidedquadratic growth restrictions. Theorem 35.1 applies for instance with F (u,∇u) =f(u) + a|∇u|m − λur|∇u|q with f of class C1, 1 < m ≤ 2, r ≥ 1 or r = 0, q > 1and λ ≥ 0. This includes in particular problem (34.3) for any p, q > 1 and µ > 0.In the special case of problem (34.3), the result was proved before in [433], [434],[498] by different techniques.

(c) As for problem (34.4), Theorem 35.1 applies when q ≥ 2, but not if 1 < q < 2,since the nonlinearity is then not Lipschitz. However, it is proved in Proposi-tion 51.16 (by different arguments) that an L∞-estimate is sufficient to preventblow-up of solutions.

(d) The growth and sign assumptions in (35.4), (35.5) are essentially optimal.Indeed, the conclusion of Theorem 35.1 fails for problem (34.5) if p > 2 (cf. Sec-tion 40; see also Section 41 and [485] for other examples). The other assumptionson F can be slightly weakened. For instance, it is enough to assume F to be C1

for ξ large.

(e) For earlier results under two-sided quadratic growth conditions on F , seee.g. [319], [469]. Note that when g = 0, the nonnegativity of u0 and the assump-tion F (0, 0) ≥ 0 are not needed. Like in [319], [469], the proof of Proposition 35.2relies on the classical Bernstein technique, which consists in applying the maxi-mum principle to the function ∂v/∂xi (or to |∇v|2), where u = φ(v). Gradientestimates can be obtained by various other techniques. Approaches based on elab-orate test-function arguments are used in [320, Theorem V.4.1 and Lemma VI.3.1],where a two-sided quadratic growth assumption is made on F (but no assump-tion on the derivatives Fu, Fξ), and in [88]. If |F | ≤ O(|ξ|m) with m < 2, resultsof this kind can be obtained via the variation-of-constants formula, or derivedfrom well-posedness results in L∞ (cf. Example 51.30, and see also [10] and [361,Lemma 5.1]). For related results in the radial case under (different) one-sidedquadratic growth assumptions, see [512]. The technique used there is still differ-ent, based on Kruzhkov’s idea of adding a new space variable. Results concerningsign-changing solutions under one-sided quadratic growth assumptions can also befound in [59], [512].

In view of the proof of Proposition 35.2, we start with a preliminary result (underweaker assumptions) which provides control of the gradient on the boundary. Theproof is based on a barrier argument (cf. [320, Lemma VI.3.1]).

Lemma 35.4. Assume that F (u, ξ) ≤ O(|ξ|2). Let T, M > 0 and let u ∈ C2,1(QT )∩ C1,0(QT ) be a solution of (34.1) satisfying (35.6). Then there holds

|∇u| ≤ C = C(M, F, Ω) on ST .

35. Well-posedness and gradient bounds 317

Proof. Let U be the solution of

−∆U = 1, 1 < |x| < 2,

U = 0, |x| = 1,

U = 1, |x| = 2.

⎫⎪⎬⎪⎭It is easily checked that

0 < U(x) < 1 and c1(|x| − 1) ≤ U(x) ≤ c2(|x| − 1), 1 < |x| < 2. (35.7)

Let x0 ∈ ∂Ω. Since Ω is uniformly smooth, there exists ρ0 ∈ (0, 1) depending onlyon Ω (independent of x0) with the following property: For any ρ ∈ (0, ρ0], thereexists y = y(ρ) ∈ R

n such that B(y, ρ) ∩ Ω = x0. Next put

V (x) = β−1 log(1 + eβMU

(x−y

ρ

)), ρ ≤ |x− y| ≤ 2ρ,

with β ≥ 1, ρ ∈ (0, ρ0]. (Observe that, up to affine changes of variables, this is justthe usual Hopf-Cole exponential transformation U = eβV .)

We want to compare u and V in the set QT = Ω× (0, T ], where

Ω = x ∈ Ω : |x− y| < 2ρ,

for suitably small ρ and y = y(ρ). Due to 0 ≤ V ≤ M + 1 and F (u, ξ) ≤ O(|ξ|2),a simple calculation shows that

−∆V ≥ β|∇V |2 + (2ρ2β)−1 ≥ F (V,∇V ), ρ < |x− y| < 2ρ, (35.8)

by taking β ≥ 1 large and then ρ ∈ (0, ρ0] small (depending only on F and M).On the other hand, using (35.6), (35.7) and imposing in addition ρ ≤ c1/2Mβ, wehave

V (x) ≥β−1eβMU

(x−y

ρ

)1 + eβMU

(x−y

ρ

) ≥ U(

x−yρ

)2β

≥ M(|x− y| − ρ) ≥ u0(x), x ∈ Ω,

and V (x) > M ≥ u(x, t) for x ∈ Ω ∩ |x − y| = 2ρ. In view of (35.8), and since∂Ω ⊂ (Ω ∩ |x − y| = 2ρ) ∪ ∂Ω, we may then apply the comparison principle inQT to deduce that

u(x, t) ≤ V (x) ≤ β−1eβMU(

x−yρ

)≤ c2(βρ)−1eβM (|x− y| − ρ), (x, t) ∈ QT ,

where we also used (35.7). Due to u(x0, t) = V (x0) = 0, it follows that

|∇u(x0, t)| = −∂u

∂ν(x0, t) ≤ −

∂V

∂ν(x0, t) ≤ c2(βρ)−1eβM , 0 < t < T.


Proof of Proposition 35.2. Consider a function φ of class C3 on some compactinterval J , with φ′ > 0 and φ(J) ⊃ [0, M ], and a constant K > 0 (φ and K willbe specified later on). Let h ∈ R

n, with |h| = 1. Given a solution u satisfying theassumptions of the proposition, we set

v := φ−1(u), w := ∂hv = h · ∇v, z(x, t) := e−Ktw.

We want to apply the maximum principle to the function z.Step 1. Derivation of the equation for z. We have

F (u,∇u) = ut −∆u = φ′(v)(vt −∆v)− φ′′(v)|∇v|2,

hence

vt −∆v =F (φ(v), φ′(v)∇v)

φ′(v)+

φ′′(v)φ′(v)

|∇v|2.

Note that z ∈ C(QT )∩L∞(QT ). Since F is C1, by differentiating the equation forv in the direction h and using Remark 48.3(i), we get z ∈W 2,1;q

loc (QT ) for all finiteq. In addition, direct computation yields

wt −∆w = a(x, t)w + b(x, t) · ∇w a.e. in QT ,

with

a = Fu +φ′′

φ′2(ξ · Fξ − F

)+

1φ′2(φ′′

φ′

)′|ξ|2 and b = Fξ + 2

φ′′

φ′2 ξ,

where F and its derivatives are evaluated at u = u(x, t), ξ = ∇u(x, t), while φ andits derivatives are evaluated at v(x, t). Setting a = a−K, we obtain

zt −∆z = a(x, t) z + b(x, t) · ∇z a.e. in QT . (35.9)

Step 2. Construction of a function φ such that a ≤ 0. Since ξ · gξ − g ≤ 0, welook for a function φ such that φ′′ ≥ 0. We take

φ(s) = eM

∫ s

0

exp(−e−λσ) dσ, s ∈ J := [0, 1],

where λ > 0 will be chosen below. For s ∈ J , we compute

φ′ = eM exp(−e−λs), φ′′ = λe−λsφ′,(φ′′

φ′)′

= −λ2e−λs.

Note that M ≤ φ′ ≤ eM , s ∈ J . In particular, we have [0, M ] ⊂ φ(J). By (35.4),(35.5), there exist a0, a1 > 0 and, for each η > 0, there exists Cη > 0, such that

Fu ≤ η|ξ|2 + Cη, ξ · Fξ − F = ξ · fξ − f + |ξ|ξ · ∂∂ξ

(g|ξ|)≤ a0|ξ|2 + a1,

36. Perturbations of the model problem: blow-up and global existence 319

for 0 ≤ u ≤ M , ξ ∈ Rn. Take λ = 2a0eM , η = 2a2

0e−λ, K ≥ Cη + λa1/M . Using

gu ≤ 0, 0 ≤ u(x, t) ≤M and 0 ≤ v(x, t) ≤ 1, it follows that, for all (x, t) ∈ QT ,

a(x, t) = Fu +λe−λv

φ′(v)

(ξ · Fξ − F − λ

φ′(v)|ξ|2)−K

≤ η|ξ|2 + Cη +λe−λv

φ′(v)

((a0 −

λ

eM)|ξ|2 + a1

)−K

≤ (η − 2a20e

−λ)|ξ|2 + Cη +λa1

M−K ≤ 0.

Step 3. Conclusion. Due to (35.6), Lemma 35.4, and φ′(v) ≥ M , we have z ≤C = C(F, Ω, M) on PT . Applying the maximum principle to equation (35.9) andusing a ≤ 0, we deduce that z ≤ C in QT . Getting back to ∂hu = eKtφ′(v)z, andsince h was arbitrary, the proposition follows.

Proof of Theorem 35.1. Assume for contradiction that T := Tmax < ∞ andlim inft→T ‖u(t)‖∞ < M for some M ∈ (0,∞). By (35.4), (35.5), there existsK > 0 such that

F (u, ξ) ≤ K(|ξ|2 + 1), 0 ≤ u ≤M + 1, ξ ∈ Rn.

Pick t0 ∈ [T − 1K , T ) such that ‖u(t0)‖∞ ≤M and let u(x, t) := M +K(t− t0) for

(x, t) ∈ Q, where Q := Ω × (t0, T ). For (x, t) ∈ Q, we have 0 ≤ u(x, t) ≤ M + 1,hence

ut −∆u− F (u,∇u) = K − F (M + K(t− t0), 0) ≥ 0.

By the comparison principle, we deduce that 0 ≤ u ≤ u ≤ M + 1 in Q. Due toProposition 35.2 it follows that supt∈(0,T ) ‖u(t)‖X <∞: a contradiction.

36. Perturbations of the model problem:blow-up and global existence

In this section, we discuss the conditions on the perturbation terms which implyor prevent blow-up.

We start with a simple criterion for equation (34.4) in bounded domains, whichis based on a modification of the eigenfunction method (see Theorem 17.1). Theidea of the proof is from [215], [330].

Theorem 36.1. Consider problem (34.4) with Ω bounded, p > 1, q ≥ 1, andu0 ∈ X+.(i) Assume p > q and set m = p/(p− q). If

∫Ω

u0ϕm1 dx > C1 = C1(Ω, p, q, a) > 0,

then Tmax(u0) < ∞.

(ii) Assume q ≥ p. Then Tmax(u0) = ∞ and supt≥0 ‖u(t)‖∞ < ∞.

Proof. (i) Denote y = y(t) :=∫Ω

u(t)ϕm1 dx. Multiplying the differential equation

in (34.4) with ϕm1 yields, for 0 < t < T := Tmax(u0),

y′ =∫

Ω

utϕm1 dx =

∫Ω

ϕm1 ∆u dx +

∫Ω

upϕm1 dx +

∫Ω

(a · ∇(ϕm1 ))uq dx.

We claim that ∫Ω

ϕm1 ∆udx ≥ −mλ1

∫Ω

uϕm1 dx. (36.1)

Since ϕm1 ∈ C2(Ω) (when 1 < m < 2), we consider (ϕ1 + ε)m and observe that

∆(ϕ1 + ε)m = m(ϕ1 + ε)m−1∆ϕ1 + m(m− 1)(ϕ1 + ε)m−2|∇ϕ1|2

≥ −mλ1(ϕ1 + ε)m−1ϕ1.

Integrating by parts, we obtain∫Ω

(ϕ1 + ε)m∆udx =∫

Ω

u∆(ϕ1 + ε)m dx +∫

∂Ω

(ϕ1 + ε)m∂νudσ

≥ −mλ1

∫Ω

u(ϕ1 + ε)m−1ϕ1 dx + εm

∫∂Ω

∂νu dσ

and (36.1) follows upon letting ε → 0.Now by Holder’s inequality we have∣∣∣∫

Ω

(a · ∇(ϕm1 ))uq dx

∣∣∣ ≤ m|a|∫

Ω

|∇ϕ1|ϕm−11 uq dx ≤ C

(∫Ω

ϕm1 up dx

)q/p

,

for some C = C(Ω, p, q, a) > 0. Combining this with Jensen’s inequality, we obtain

y′ ≥∫

Ω

upϕm1 dx− C

(∫Ω

upϕm1 dx

)1/p

− C(∫

Ω

upϕm1 dx

)q/p

≥ 12yp − C,

for some C = C(Ω, p, q, a) > 0. We infer that u cannot exist globally whenevery(0) > (2C)1/p.

(ii) Without loss of generality, we may assume that a = |a|e1 and that Ω ⊂x ∈ R

n : 0 < x1 < L for some L > 0. We seek for a (stationary) supersolutionof (34.4) of the form v(x) = Keαx1, for arbitrarily large K > 0 (to guaranteeK ≥ ‖u0‖∞) and some α > 0. The condition to ensure this is thus

−α2Keαx1 ≥ Kpeαpx1 − |a|αqKqeαqx1 , 0 < x1 < L,

which is satisfied if

αq|a|Kq−peα(q−p)x1 ≥ 1 + α2K1−peα(1−p)x1 , 0 < x1 < L.

Since q ≥ p > 1, it is thus sufficient that αq|a|Kq−p ≥ 1 + α2K1−p. This is truefor α = 2/q|a| and all large K > 1. It then follows from the comparison principlein Proposition 52.16, that 0 ≤ u(x, t) ≤ v(x) ≤ KeαL in Ω, as long as u(t) exists.By Proposition 51.16, this implies global existence.

We now turn to problem (34.3) (in bounded and unbounded domains). Webegin with a result from [497] which shows that finite-time blow-up occurs forlarge initial data when p > q.

Theorem 36.2. Consider problem (34.3) with p > q > 1, µ > 0. Let u0 = λφ,with φ ∈ X+, φ ≡ 0, λ > 0. If λ is sufficiently large, then Tmax(u0) <∞.

Remarks 36.3. (i) For problem (34.3) the eigenfunction method does not seemto apply, and the proof of Theorem 36.2 relies on a different technique, based onself-similar blowing-up subsolutions. For earlier results in that direction (and othermethods), see for instance [129], [304], [433].

(ii) When p > q, by Young’s inequality, we have

|a · ∇(uq)| ≤ q|a|uq−1|∇u| ≤ 12up + µ|∇u|m, m = p/(p− q + 1) < p,

for some µ = µ(a, p, q) > 0, so that any solution of (34.4) is a supersolution ofut − ∆u = 1

2up − µ|∇u|m. Consequently, Theorem 36.2 implies blow-up of thesolution of (34.4) for large initial data. However, the criterion in Theorem 36.1(i)is more precise.

(iii) Blow-up for slow decay initial data. For problems (34.3) with p > q ≥2p/(p+ 1), and (34.4) with q = (p +1)/2, there are blow-up results for slow decayinitial data in Ω = R

n, similar to those known for the model problem (15.1). In fact,the conclusion of Theorem 17.12 remains valid in this case, with a different constanton the RHS of (17.14) [497]. The proof is based on Theorem 36.2, rescaling andcomparison arguments. Such results extend to more general unbounded domains(containing a cone or a paraboloid); see [497], [461].

Proof of Theorem 36.2. We seek a (self-similar) subsolution of (34.3) of theform:

v(x, t) =1

(1− εt)kV

(|x|

(1− εt)m

), t0 ≤ t < 1/ε,

where V is defined by

V (y) = 1 +A

2− y2

2A, y ≥ 0.

Here A, k, m, t0, ε > 0 (with t0 < 1/ε) are to be determined. Set R =(A(2+A)

)1/2,so that V (R) = 0. Note that v(x, t) > 0 if and only if (x, t) ∈ D, where

D :=(x, t) : t0 ≤ t < 1/ε, |x| < R(1− εt)m

,

and that v is smooth in D. We will verify that Pv := vt −∆v − vp + µ|∇v|q ≤ 0in D. We compute, setting y = |x|/(1− εt)m for convenience:

Pv =ε(kV (y) + myV ′(y))

(1− εt)k+1−

V ′′(y) + n−1y V ′(y)

(1− εt)k+2m− V p(y)

(1− εt)kp+ µ

|V ′(y)|q(1− εt)(k+m)q

.

The function V obviously satisfies

1 ≤ V (y) ≤ 1 + A/2, −1 ≤ V ′(y) ≤ 0, for 0 ≤ y ≤ A,

0 ≤ V (y) ≤ 1, −R/A ≤ V ′(y) ≤ −1, for A ≤ y ≤ R,

V ′′(y) + (n− 1)V ′(y)/y = −n/A, for 0 < y < R.

We first choose

k =1

p− 1, 0 < m < min

12,

p− q

q(p− 1)

,

so that kp = k + 1 > k + 2m and k + 1 > (k + m)q, and next we choose:

A > k/m, ε <1

k(1 + A/2).

In the case 0 ≤ y ≤ A, by using also V ′ ≤ 0 and by taking t0 = t0(ε, k, m, q, A, n, µ)sufficiently close to 1/ε, we obtain

Pv(x, t) ≤ εk(1 + A/2)− 1(1 − εt)k+1

+n/A

(1− εt)k+2m+

µ

(1− εt)(k+m)q

≤ (1− εt)−k−1(εk(1 + A

2 )− 1 + nA (1− εt0)1−2m

+ µ(1− εt0)k+1−(k+m)q)≤ 0.

In the case A ≤ y < R, by taking t0 = t0(ε, k, m, q, A, n, µ) still closer to 1/ε, weget

Pv(x, t) ≤ ε(k −mA)(1 − εt)k+1

+n/A

(1− εt)k+2m+

µ(R/A)q

(1 − εt)(k+m)q

≤ (1 − εt)−k−1(ε(k −mA) + n

A (1− εt0)1−2m

+ µ(

RA

)q(1 − εt0)k+1−(k+m)q)≤ 0.

Now, by translation, one can assume without loss of generality that 0 ∈ Ω andφ ≥ C in B(0, ρ) for some ρ, C > 0. Therefore, for t0 close to 1/ε and λ > 0 largeenough, we have u0 ≥ v(·, t0) in B(0, R(1− εt0)m), hence in Ω. Moreover, we have

v ≤ 0 on ∂Ω × (t0, 1/ε). If Tmax(u0) ≥ 1/ε − t0, it follows from the comparisonprinciple that

u(x, t− t0) ≥ v(x, t) in D.

Since v(0, t)→∞ as t→ 1/ε, we conclude that Tmax(u0) ≤ 1/ε− t0 <∞.

The next result from [498], [481] shows in particular that the blow-up conditionp > q in Theorem 36.2 is optimal for bounded domains (see [185], [434] for ear-lier results in that direction). However, for general unbounded domains, the issuedepends in a crucial way on the geometry of the domain, through the notion ofinradius ρ(Ω) (cf. Section 19 and Appendix D).

Theorem 36.4. Consider problem (34.3) with q ≥ p > 1, µ > 0.(i) Assume ρ(Ω) < ∞. Then for all u0 ∈ X+, there holds Tmax(u0) =∞ and

supt≥0

‖u(t)‖∞ < ∞.

Assume in addition that u0 ∈ W 1,r0 (Ω) for some finite r > n max(1, q − 1). There

exist µ0, λ > 0 (depending only on Ω, p, q, r) such that, if µ ≥ µ0, then

‖u(t)‖s ≤ C(u0) e−λt, t ≥ 0, r ≤ s ≤ ∞. (36.2)

(ii) Assume ρ(Ω) = ∞. Then there exists u0 ∈ X+, such that either

Tmax(u0) < ∞, or Tmax(u0) = ∞ and limt→∞ ‖u(t)‖∞ =∞.

Furthermore, u0 can be taken in W 1,r0 (Ω) for r large.

We start with assertion (i). The proof of global existence and boundedness isbased on comparison arguments. The idea is to construct a stationary supersolu-tion v in the exterior of a ball of small radius ε, which is radial and whose minimumis larger than ‖u0‖∞. The solution u is thus dominated by all the translates ofv, centered at points y such that B(y, ε) ⊂ Ωc (these supersolutions play the roleof a barrier). Since ρ(Ω) < ∞ and Ω is uniformly regular, any point x of Ω is atbounded distance of such a point y. This guarantees a uniform bound for u, henceglobal existence in view of the gradient estimates in Section 35. The decay willbe proved by a multiplier argument, using multiplication by a power of u and thePoincare inequalities (which are valid due to ρ(Ω) <∞).

Remark 36.5. Although the comparison function v below is unbounded, v and∇v are bounded on the set (x, t) ∈ Ω× [0, T ] : u > v for each T < Tmax(u0), dueto u ∈ L∞(QT ). Consequently the comparison principle can be applied in view ofRemark 52.11(i).

Proof of Theorem 36.4(i). Applying the finiteness assumption on ρ(Ω) andthe uniform regularity of Ω, we may choose ε ∈ (0, 1) such that for any ball B of

radius ρ(Ω) + 1, B ∩ Ωc contains a ball of radius ε. Let a be a fixed point in Ω,and let us pick xa such that

B(xa, ε) ⊂ Ωc

and|xa − a| ≤ ρ(Ω) + 1. (36.3)

We seek for a supersolution of (34.3) of the form v(x, t) = Keαr, r = |x − xa|,α ≥ 0. The inequality Pv := vt −∆v + µ|∇v|q − vp ≥ 0 needs to be checked onlyfor r ≥ ε. The condition to ensure is thus

−α2Keαr − αn− 1

rKeαr + µαqKqeαqr −Kpeαpr ≥ 0, r > ε,

which is satisfied if

µαqKq−1eα(q−1)r ≥ Kp−1eα(p−1)r + α2 + αn− 1

ε, r > ε.

Since q ≥ p > 1, this is achieved whenever

µαqKq−1 ≥ 2Kp−1 and µαqKq−1 ≥ 2α2 + 2αn− 1

ε.

It thus suffices to choose α = (2/µ)1/q and next

K = max‖u0‖∞, 1,

(α2 + α(n− 1)/ε

)1/(q−1).

It then follows from the comparison principle that 0 ≤ u(x, t) ≤ v(x, t) in Ω, aslong as u(t) exists. In particular, using (36.3), we have

0 ≤ u(a, t) ≤ K exp[(2/µ)1/q(ρ(Ω) + 1)].

Since a was an arbitrary point in Ω, we deduce that u(t) remains bounded in L∞

on its existence interval. By virtue of Theorem 35.1, this implies global existence.Let us next prove the exponential decay statement. Since we now assume that

u0 ∈ W 1,r0 (Ω), it follows from Example 51.29 that u ∈ C([0,∞), W 1,r

0 (Ω)) ∩C((0,∞), W 2,r ∩W 1,r

0 (Ω))∩C1((0,∞), Lr(Ω)). We multiply the equation by ur−1

and integrate over Ω, which yields, for t > 0,

1r

d

dt

∫Ω

ur dx =∫

Ω

ur−1∆udx +∫

Ω

up+r−1 dx− µ

∫Ω

ur−1|∇u|q dx.

Integrating by parts, it follows that6

1r

d

dt

∫Ω

ur dx =∫

Ω

up+r−1 dx− (r − 1)∫

Ω

ur−2|∇u|2 dx− µ

∫Ω

ur−1|∇u|q dx,

=∫

Ω

up+r−1 dx− C1

∫Ω

|∇ur/2|2 dx− µC2

∫Ω

∣∣∇(u q+r−1q)∣∣q dx.

6Note that we have r > 2 if n ≥ 2, thus integration by parts can be carried out withoutdifficulty. If n = 1 and 1 < r < 2, this can still be done easily.


Here and in what follows, C, C1, C2 denote any constant depending only on p, q, rand Ω, but not on µ.

Now, due to Proposition 50.1, we may apply the Poincare inequality in H10 (Ω)

and in W 1,q0 (Ω) to get

1r

d

dt

∫Ω

ur dx ≤∫

Ω

up+r−1 dx− C

∫Ω

ur dx− µC

∫Ω

uq+r−1 dx. (36.4)

Using the inequality

xp+r−1 ≤ εxr + C(p, q)ε−(q−p)/(p−1)xq+r−1, x ≥ 0, ε > 0

in case q > p, it follows from (36.4) that

d

dt

∫Ω

ur dx ≤ −C

∫Ω

ur dx

whenever q ≥ p and µ > µ0(Ω, p, q, r) large enough. Consequently,∫Ω

ur(t) dx ≤ exp(−Ct)∫

Ω

ur0 dx, t > 0. (36.5)

To prove exponential decay in L∞, we use an argument of comparison with themodel problem (15.1). Fix t0 > 0. By (36.5), we have ‖u(t0)‖r ≤ M := ‖u0‖r.Therefore, since r > n(p − 1)/2, by Theorem 15.2, the solution v of (15.1) withinitial data v(0) = u(t0) exists on a time interval [0, τ ] with τ = τ(M) (independentof t0) and satisfies ‖v(t)‖∞ ≤ C‖v(0)‖r t−n/2r on (0, τ ]. Since u(t0 + t) ≤ v(t) on[0, τ ] by the comparison principle, it follows from (36.5) that

‖u(t)‖∞ ≤ C‖u(t− τ)‖r τ−n/2r ≤ C(M) exp(−C (t− τ)), t ≥ τ,

hence (36.2) with s = ∞. The general case r ≤ s ≤ ∞ follows by interpolatingbetween s = r and s = ∞.

The main ingredient of the proof of Theorem 36.4(ii) (and of Theorem 36.7below) is the following lemma.

Lemma 36.6. Let p > 1, q > 2p/(p + 1) and µ ≥ 0. There exist η, ε, R > 0 anda (radial) function v ≥ 0, of class C2 on R

n × R+, satisfying:

Pµv := vt −∆v − vp + µ|∇v|q ≤ 0, x ∈ Rn, t ≥ 0, (36.6)

supp (v(t)) ⊂ B(0, R + ηt), t ≥ 0, (36.7)

‖v(t)‖∞ = v(0, t) ≥ εt, t ≥ 0, (36.8)


limt→∞ v(x, t) =∞, x ∈ R

n, (36.9)

vt(x, t) ≥ 0, x ∈ Rn, t ≥ 0, (36.10)

and‖∇v‖L∞(Rn×R+) ≤ 1. (36.11)

Intuitively, the idea is to seek an unbounded global subsolution, whose gradientremains uniformly bounded, so that the damping effect of the gradient term cannever become too important even for large q. This subsolution will take the formof a spherical “expanding wave”, which propagates radially away from the originwith an increasing maximum at 0.

Proof of Lemma 36.6. We need two auxiliary functions. Let us first define afunction f : R → R, of class C2, by

f(s) =

⎧⎨⎩0, s ≤ 0,

4s3(1− s), 0 ≤ s ≤ 1/2,

s− 1/4, s ≥ 1/2.

It is easily seen that f satisfies, for some ε > 0,

0 ≤ f ′ ≤ 1, f ′′ ≥ 0, s ∈ R,

f ′′ + fp ≥ 3εf ′, s ≤ 1/2 and fp ≥ 3εf ′, s ≥ 1/2.

Next, we define β : R+ → R, as

β(s) =

⎧⎨⎩ s +(M − s)3

3M2, 0 ≤ s ≤M,

s, s > M,

with M = 2n/ε. The function β is of class C2 on R+, with the following properties:

0 ≤ β(s) ≤ M, 0 ≤ s ≤ M,

s ≤ β(s), 0 ≤ β′ ≤ 1, 0 ≤ β′′ ≤ ε/n, s ∈ R+,

β(0) = M/3, β′(0) = 0.

Now we set

U(x, t) = f(M +

12

+ εt− β(|x|)), x ∈ R

n, t ≥ 0,

which is of class C2 on Rn×R+. We compute (omitting the argument in f , f ′, f ′′

for simplicity):

∇U = − x

|x|β′(|x|)f ′ (0 if x = 0), ∆U = β′2(|x|)f ′′ − f ′∆

(β(|x|)

),


∆(β(|x|)

)= β′′(|x|) +

n− 1|x| β′(|x|) ≤ n sup β′′ ≤ ε.

First taking µ = ε in (34.3), we have

PεU = εf ′ − β′2(|x|)f ′′ + ∆β(|x|)f ′ − fp + ε|β′(|x|)f ′|q ≤ 3εf ′ − β′2(|x|)f ′′ − fp.

If s = 1/2 + M + εt − β(|x|) ≥ 1/2, then fp ≥ 3εf ′ hence PεU(x, t) ≤ 0. Onthe other hand, if s ≤ 1/2, then β(|x|) ≥ M + εt ≥ M . Hence β′(|x|) = 1 andPεU(x, t) ≤ 3εf ′ − f ′′ − fp ≤ 0. Now, for a general µ > 0, replacing U by

Uα(x, t) = α2/(p−1)U(αx, α2t),

we get

PµUα = α2p/(p−1)[Ut −∆U − Up + µα(q(p+1)−2p)/(p−1)|∇U |q

](αx, α2t)

≤ α2p/(p−1)[PεU ](αx, α2t) ≤ 0,

for α > 0 sufficiently small since q > 2p/(p + 1), which proves (36.6) with v = Uα.Finally, (36.7)–(36.11) are straightforward consequences of the definition of f (takeR = (M + 1/2)/α and η = εα and replace ε in (36.8) by εα2p/(p−1)).

Proof of Theorem 36.4(ii). Let Rj be a sequence of positive reals, Rj → ∞.From the hypotheses, there is a sequence of disjoint balls Bj = B(xj , R

′j) ⊂ Ω,

with R′j > Rj . We are going to construct a suitable subsolution w = w(x, t) of

(34.3) on Ω by taking advantage of the scaling properties of the equation. With vas in Lemma 36.6, we set:

wj(x, t) =1

j2/(p−1)v(x− xj

j,γj(t)j2

), x ∈ R

n, t ≥ 0, j ∈ N∗,

with γj(t) = Mjt/(Mj + t), where the constants Mj > 0 will be adjusted later. By(ii)–(iii) in Lemma 36.6, we have:

supp (wj(t)) ⊂ B(xj , j(R + ηMj/j2)

), t ≥ 0,

‖wj(t)‖∞ ≥ εγj(t)j2p/(p−1)

→ εMj

j2p/(p−1)as t→∞.

For (x, t) ∈ Rn × R+, it follows from (i) that

Pwj = j−2p/(p−1)[γ′

j(t)vt −∆v − vp + µj(2p−q(p+1))/(p−1)|∇v|q](x− xj

j,γj(t)j2

)≤ j−2p/(p−1)

[vt −∆v − vp + µ|∇v|q

](x− xj

j,γj(t)j2

)≤ 0,


where we have used the fact that q ≥ p > 2p/(p + 1), vt ≥ 0 and γ′j(t) =

M2j /(Mj + t)2 ≤ 1. We now choose

Mj = j1+2p/(p−1) and Rj = j(R + ηMj/j2)

and define the function w as:w =

∑j≥1

wj .

Note that each wj is supported on Bj and that the Bj are disjoint. By Lemma 36.6,it is clear that w is C2 on R

n ×R+, and hence is a classical subsolution of (34.3).Moreover, by the choice of γj , w is bounded on R

n × [0, T ] for each T > 0. Wenote that w(0) ∈ X+. (Also, since

‖wj(0)‖∞ ≤ j−2/(p−1)‖v(0)‖∞ and ‖∇wj(0)‖∞ ≤ j−(p+1)/(p−1)‖∇v(0)‖∞,

it follows from the choice of Rj that w(0) ∈ W 1,r0 (Ω) for all large r.) By the

comparison principle, the solution of (34.3) with initial data w(0) remains abovew(t) as long as it exists, which implies the desired conclusion.

It is not known whether blow-up may occur in finite time when q ≥ p and ρ(Ω) =∞ (except, of course, for the trivial example when u solves the corresponding ODE,i.e. u0(x) = C in Ω = R

n). The next result from [498] shows that infinite-timeblow-up can occur in the case Ω = R

n.

Theorem 36.7. Consider problem (34.3) with q ≥ p > 1, µ > 0 and Ω = Rn.

(i) Assume that u0 ∈ X+ has compact support. Then Tmax(u0) = ∞.

(ii) There exists u0 ∈ X+ with compact support, such that Tmax(u0) = ∞ and u isunbounded. Actually, it even holds

limt→∞u(x, t) =∞, for all x ∈ R

n.

Proof. (i) We shall actually prove that the following exponential decay condition(instead of compact support) is sufficient for global existence:

0 ≤ u0(x) ≤ Ce−ε|x.a|, x ∈ Ω, for some C > 0, a ∈ Rn, |a| = 1,

where ε is any positive number if q > p, or ε = µ−1/p if q = p. Without loss ofgenerality, we may assume that a is the unit vector in the x1-direction. We claimthat, for a suitable choice of α, the functions

v±(x, t) = C exp(αt± εx1)

are (traveling wave) supersolutions. If q > p and β > 0, or if q = p and β = 1, wehave the elementary inequality

xq ≥ β(q−p)/(q−1)xp − βx, x ≥ 0.

Therefore,

∂tv± −∆v± + µ|∇v±|q − vp±

≥ ∂tv± −∆v± + µβ(q−p)/(q−1)|∇v±|p − µβ|∇v±| − vp±

= C exp(αt± εx1)(α− ε2 − µβε

)+ Cp exp

[p(αt± εx1)

] (µβ(q−p)/(q−1)εp − 1

).

It thus suffices to choose β = (µεp)−(q−1)/(q−p) and α = ε2 + µβε, if q > p andε > 0, or β = 1 and α = ε2 + µε, if q = p and ε = µ−1/p. Then we get, thanks tothe comparison principle

0 ≤ u(x, t) ≤ v±(x, t), x ∈ Rn, 0 ≤ t < T,

where T = Tmax(u0) (note that the comparison principle applies, for the samereason as in Remark 36.5). Consequently,

0 ≤ u(x, t) ≤ C exp(αt− ε|x1|), x ∈ Rn, 0 ≤ t < T,

hence in particular‖u(t)‖∞ ≤ C exp(αt), 0 ≤ t < T.

By virtue of Theorem 35.1, this implies global existence.(ii) Taking u0 = v(0), with v as in Lemma 36.6, it is an immediate consequence

of that lemma and part (i).

Remarks 36.8. (i) Blow-up set. In Theorem 36.7(ii), we have global blow-up(in infinite time), i.e. the blow-up set is the whole of R

n. Infinite-time blow-up forq ≥ p is also known to occur when Ω is a cone (see [498]). But in this case, blow-uptakes place only at infinity (the solution remaining bounded for t ≥ 0 in compactsubsets).

(ii) In Theorem 36.4, the largeness assumption on µ0 for decay is necessary ingeneral. Indeed, when q > 2p/(p + 1), p < pS and Ω is a ball, there exist positivestationary solutions (see [129, Corollary 5.4]).

(iii) For problem (34.3) where the gradient term is replaced with −µur|∇u|q,related results can be found in [497], [58].

37. Fujita-type results

We consider the Cauchy problems associated with (34.3) and (34.4), i.e.:

ut −∆u = up − µ|∇u|q, x ∈ Rn, t > 0,

u(x, 0) = u0(x), x ∈ Rn,

(37.1)

andut −∆u = up − a · ∇(uq), x ∈ R

n, t > 0,

u(x, 0) = u0(x), x ∈ Rn.

(37.2)

In this section, we give Fujita-type results for problems (37.1) and (37.2), i.e. wefind conditions which guarantee that the solution blows up in finite time for allu0 ≥ 0, u0 ≡ 0 (and not only for large initial data as in the previous section).

For (37.1), the following result is due to [373] and is based on a method ofrescaled test-functions.

Theorem 37.1. Consider problem (37.1) with p > 1, q = 2p/(p+1). There existsµ0(n, p) > 0 such that if

p < 1 +2n

and µ ≤ µ0,

then Tmax(u0) < ∞ for any nontrivial u0 ∈ X+.

We can complement Theorem 37.1 with the following result.

Theorem 37.2. Consider problem (37.1) with p, q > 1, and assume that at leastone of the following assumptions holds:

p > 1 +2n

;(i)

q <2p

p + 1;(ii)

q =2p

p + 1and µ > µ1(n, p) > 0 large enough.(iii)

Then Tmax(u0) = ∞ and supt≥0 ‖u(t)‖∞ < ∞ for some nontrivial u0 ∈ X+.

Remarks 37.3. (i) Critical exponents. The value q = p is critical for the blow-up and global existence properties of equation (34.3), as shown in the previoussection. Another particular role is played by q = 2p/(p + 1). Indeed, for thisvalue of q, the differential equation in (34.3) enjoys the same scaling propertiesas for µ = 0. Namely, for any solution u and any α > 0, the rescaled functionuα(x, t) := α2/(p−1)u(αx, α2t) is still a solution. This property is reflected in theexistence of blowing-up self-similar solutions (cf. Remark 39.8(i)).

(ii) It seems to be unknown whether (37.1) admits any global solutions when2p/(p + 1) < q 2p/(p + 1) and p ≤ n/(n− 2)+ [470].

(iii) A stronger result than Theorem 37.1 actually holds: Under the assumptionsof that theorem there exist no nontrivial nonnegative distributional solutions ofut −∆u = up − µ|∇u|q in Q = R

n × (0,∞), with u ∈ Lploc(Q) and ∇u ∈ L2

loc(Q).This follows from a small modification of the proof below and from similar argu-ments as in Step 1 of the proof of Theorem 18.1(i).

Proof of Theorem 37.1. Assume that u ≥ 0 is a global solution of (37.1),classical for t > 0, with u ∈ L∞

loc(Rn × [0,∞)).

Step 1. Let α ∈ (0, 1), a = (p− α)/(p− 1) > 1 and Ai > 0, i = 1, . . . , 4, with

C1 := α−A1 − µA3 ≥ 0. (37.3)

For simplicity, we shall write∫

for the space integral∫

Rn and∫ ∫

for the time-spaceintegral

∫∞0

∫Rn . We claim that for any compactly supported ϕ ∈ C1(Rn× [0,∞)),

ϕ ≥ 0, there holds

C2

∫ ∫up−αϕ ≤ C3

∫ ∫|∇ϕ|2aϕ1−2a + C4

∫ ∫|ϕt|aϕ1−a, (37.4)

where

C2 = 1− C(p, α)(A−1

1 A1−a′2 + µA−p

3 + A4

), C3 = A2/4A1, C4 = C(p, α)A1−a

4

(the function ϕ will be later chosen such that the integrals on the RHS will befinite).

Fix τ, ε > 0 and put uε = u + ε. Multiplying the equation by u−αε ϕ and inte-

grating by parts, we get∫ ∞

τ

∫upu−α

ε ϕ + α

∫ ∞

τ

∫|∇u|2u−1−α

ε ϕ +1

1− α

∫u1−α

ε (·, τ)ϕ(·, τ)

=∫ ∞

τ

∫u−α

ε ∇u · ∇ϕ + µ

∫ ∞

τ

∫|∇u|qu−α

ε ϕ +1

α− 1

∫ ∞

τ

∫u1−α

ε ϕt

=: I1 + µI2 + I3.(37.5)

Let us estimate I1, I2, I3 in terms of the double integrals appearing on the LHS.Repeatedly using Young’s inequality, we obtain

I1 ≤ A1

∫ ∞

τ

∫|∇u|2u−1−α

ε ϕ + B1

∫ ∞

τ

∫|∇ϕ|2u1−α

ε ϕ−1

≤ A1

∫ ∞

τ

∫|∇u|2u−1−α

ε ϕ + B1A2

∫ ∞

τ

∫|∇ϕ|2aϕ1−2a + B1B2

∫ ∞

τ

∫up−α

ε ϕ,

I2 ≤ A3

∫ ∞

τ

∫|∇u|2u−1−α

ε ϕ + B3

∫ ∞

τ

∫up−α

ε ϕ

andI3 ≤ A4

∫ ∞

τ

∫up−α

ε ϕ + C4

∫ ∞

τ

∫|ϕt|aϕ1−a,

where B1 = (4A1)−1, B2 = C(p, α)A1−a′2 and B3 = C(p)A−p

3 . Plugging the aboveestimates in (37.5), we find that∫ ∞

τ

∫upu−α

ε ϕ− (B1B2 + µB3 + A4)∫ ∞

τ

∫up−α

ε ϕ + C1

∫ ∞

τ

∫|∇u|2u−1−α

ε ϕ

≤ C3

∫ ∞

τ

∫|∇ϕ|2aϕ1−2a + C4

∫ ∞

τ

∫|ϕt|aϕ1−a.

Due to assumption (37.3), the third term in the LHS can be left out. Since ϕis compactly supported, with the help of the monotone convergence theorem, wemay pass to the limit ε → 0, and then τ → 0, in the first two terms of the LHS.This yields (37.4).

Step 2. Choose0 < α < 1− n(p− 1)/2, (37.6)

A1 = α/2 and A3 = 1. By taking A2 large, A4 small (depending only on n, p), andthen µ < µ0(n, p) small, we have (37.3) and C2 > 0.

Now consider ϕ of the form ϕ(x, t) = ψ( |x|

R

)ψ(

tR2

). Here R > 0, ψ ∈ C1([0,∞))

satisfies ψ′ ≤ 0 and

ψ(s) =

⎧⎨⎩1, 0 ≤ s ≤ 1,

(2− s)m, 3/2 ≤ s ≤ 2,

0, s ≥ 2,

with m > 2a > 2. Inequality (37.4) implies

C2

∫ ∫Σ

up−α ≤ C3

∫ ∫Σ′|∇ϕ|2aϕ1−2a + C4

∫ ∫Σ′|ϕt|aϕ1−a, (37.7)

where

Σ =(x, t) : |x| ≤ R, 0 ≤ t ≤ R2

, Σ′ =

(x, t) : |x| ≤ 2R, 0 ≤ t ≤ 2R2

.

Observe that the integrals on the RHS are finite (the integrands are continuous,including at |x| = 2R, t = 2R2 due to m > 2a). The substitutions x = Ry, t = R2sinto the integrals on the right-hand side of (37.7) then yield

C2

∫ ∫Σ

up−α ≤ CRn+2−2a.

Since n+2−2a < 0 due to (37.6), by letting R →∞, we conclude that u ≡ 0.

Proof of Theorem 37.2. By virtue of Theorem 35.1, it suffices to obtain auniform estimate of u.

If p > 1 + (2/n), then u is a subsolution of the same problem with µ = 0 andthe same initial data. Global existence for small initial data then follows fromTheorem 20.1 in view of the comparison principle.

If q = 2p/(p + 1) and µ > µ1(p) large enough we shall show that there existsa (bounded stationary) supersolution of the form U(x) = ε(1 + |x|2)−a, witha = 1/(p− 1), which will imply the desired conclusion.

We have

∇U = −2εax(1 + |x|2)−(a+1), −∆U = 2εa[n + (n− 2− 2a)|x|2

](1 + |x|2)−(a+2).

By choosing 0 < ε, r0 < 1 small enough (depending only on n, p), we first guaranteethat

−∆U ≥ anε ≥ εp ≥ Up, |x| ≤ r0. (37.8)

Next, for |x| > r0, there holds

∆U ≤ C1(1 + |x|2)−(a+1) = C1(1 + |x|2)−p/(p−1), Up ≤ (1 + |x|2)−p/(p−1)

and|∇U |q ≥ C2(1 + |x|2)−q(a+(1/2)) = C2(1 + |x|2)−p/(p−1)

for some C1, C2 > 0 depending only on n, p. Therefore,

−∆U + µ1|∇U |q ≥ Up, |x| > r0,

provided µ1 = µ1(n, p) is chosen large enough. This along with (37.8) guaranteesthat U is a supersolution.

Finally, if q < 2p/(p + 1) and µ > 0, let us put V (x) = α2/(p−1)U(αx). Since|∇U | is bounded, we have |∇U |q ≥ c|∇U |2p/(p+1) in R

n for some c > 0. For α > 0sufficiently small, it follows that(

−∆V + µ|∇V |q − V p)(x)

= α2p/(p−1)(−∆U + µα(q(p+1)−2p)/(p−1)|∇U |q − Up

)(αx)

≥ α2p/(p−1)(−∆U + µ1|∇U |2p/(p+1) − Up

)(αx) ≥ 0,

so that V is a supersolution.

We now turn to problem (37.2). The following result from [5] shows that thecritical number (for p) may depend on both n and q.

Theorem 37.4. Consider problem (37.2) with p, q > 1, a = 0, and set

p1 = p1(n, q) := min(1 +

2n

, 1 +2q

n + 1

).

(i) If q ≤ p ≤ p1, then Tmax(u0) < ∞ for any nontrivial u0 ∈ X+.(ii) If p > p1, then Tmax(u0) =∞ for some nontrivial u0 ∈ X+.

Remarks 37.5. (a) Critical exponents. It was also shown in [5] that whenq = 1, the critical exponent becomes p = 1+2/n. We thus observe that the criticalexponent p1(n, q) is a discontinuous function of q (since p1(n, q)→ 1 + 2/(n + 1),as q → 1+, in view of Theorem 37.4).

(b) It is known (see [498, Proposition 3.6] and its proof) that blow-up in finite orinfinite time can occur for (37.2) whenever q ≥ p > 1 and that this actually occursfor all nontrivial u0 ≥ 0 when q > p > 1 and p < 1 + 2/n (see [498, Remark 3.4]and [5]). However it is unknown whether the blow-up time is finite or infinite.

(c) The following proof is a simplification of the original proof of [5] (especiallyfor part (ii) in Case 1 below). Moreover, it yields uniform decay rates for suitablysmall data in assertion (ii).

Proof of Theorem 37.4. (i) We shall prove the result only for p < p1, theequality case being more involved.

Setφ(x) = C exp

(− 1

n(1 + |x|2)1/2

),

where C > 0 is chosen so that∫

Rn φ(x) dx = 1. For i = 1, . . . , n, we have

|∂xiφ| ≤φ

n, ∂2

xixiφ ≥ −φ

n. (37.9)

Without loss of generality, we may assume that a = |a|e1. Let γ ≥ 0 to be fixedbelow. Let λ ∈ (0, 1], and put φλ(x) = λn+γφ(λ1+γx1, λx′), where x = (x1, x

′).By (37.9), we have∫

Rn

φλ(x) dx = 1, ∆φλ ≥ −λ2φλ, |(φλ)x1 | ≤ λ1+γφλ.

Multiplying equation (37.2) by φλ and integrating on Rn, we obtain, for t > 0,

d

dt

∫Rn

uφλ =∫

Rn

φλ∆u +∫

Rn

upφλ − |a|∫

Rn

(uq)x1φλ

=∫

Rn

u∆φλ +∫

Rn

upφλ + |a|∫

Rn

uq(φλ)x1

≥ −λ2

∫Rn

uφλ +∫

Rn

upφλ − |a|λ1+γ

∫Rn

uqφλ

(this can be easily justified by using the exponential decay of φ and the fact thatu(·, t) ∈ BC2(Rn)).

Denote yλ(t) =∫

Rn u(t)φλ. If q < p, by Young’s inequality, we observe that

|a|λ1+γuq = up(q−1)/(p−1)(|a|λ1+γu(p−q)/(p−1))

≤ 12up + Cλ(1+γ)(p−1)/(p−q)u

(37.10)

for some C = C(p, q, |a|) > 0. If q = p, then (37.10) is obviously true with C = 0for all λ small. Using

∫Rn upφλ ≥ yp

λ (owing to Jensen’s inequality) and (37.10),we deduce that

y′λ(t) ≥ 1

2ypλ − (λ2 + Cλ(1+γ)(p−1)/(p−q))yλ.

It follows that yλ, and hence u, cannot exist for all t > 0 whenever the RHS inthe previous inequality is positive at t = 0. This is satisfied if(∫

Rn

u0(x)φ(λ1+γx1, λx′) dx)p−1

> 2λ−(n+γ)(p−1)(λ2 + Cλ(1+γ)(p−1)/(p−q)).

(37.11)Now, since p < p1, by choosing 0 < γ < γ+ := 2/(p− 1)− n close to γ+, we get

(n + γ)(p− 1) < 2 and n + γ < (1 + γ)/(p− q).

Since, by monotone convergence, the LHS in (37.11) converges to(φ(0)

∫Rn u0

)p−1

∈ (0,∞] as λ → 0, (37.11) holds for λ > 0 sufficiently small and we conclude thatTmax(u0) <∞.

(ii) By Proposition 51.16, it suffices to obtain a uniform estimate of u onbounded time intervals.

Case 1: q > 1 + (1/n). This case is simple, since one can directly build a (self-similar) supersolution of (37.2) under the form

v(x, t) = tαG(x, t)

for some 0 < α < n/2, where G = (4π)n/2G and G is the Gaussian heat kernel.Indeed, setting k = n/2− α, the function v satisfies

vt −∆v − vp + a · ∇(vq) = tα(Gt −∆G) + αtα−1G− tαpGp + tαqa · ∇(Gq)

= αt−k−1e−|x|2/4t − t−kpe−p|x|2/4t

− qt−kq−1/2(x · a

2√

t

)e−q|x|2/4t

≥(αt−k−1 − t−kp − Ct−kq−1/2

)e−|x|2/4t,

where we used se−qs2 ≤ Ce−s2, s ≥ 0. Now, since p > p1 = 1+2/n and q > 1+1/n,

by taking α > 0 sufficiently small, it follows that kp > k + 1 and kq + 1/2 > k + 1,so that vt−∆v−vp +a ·∇(vq) ≥ 0 in R

n for t ≥ t0, where t0 ≥ 1 is large enough. Ifu0(x) ≤ t

−n/20 exp(−|x|2/4t0), the comparison principle in Proposition 52.16 then

guarantees that u(t) ≤ v(t0 + t) on [0, Tmax(u0)) and u exists globally.Case 2: q ≤ 1+(1/n). This case is more involved and requires the consideration

of the auxiliary problem:vt −∆v = −(1 + t)ra · ∇(vq), t > 0, x ∈ R

n,

v(x, 0) = u0(x), x ∈ Rn,

(37.12)

with r > 0. By Proposition 51.16, for any u0 ∈ X+, problem (37.12) has a uniqueclassical solution v ≥ 0. By the maximum principle, we have

‖v(t)‖∞ ≤ ‖u0‖∞, (37.13)

which guarantees the global existence of v, in view of (51.39). Moreover, v satisfies

v ∈ L∞loc((0,∞), BC2(Rn)). (37.14)

If in addition u0 ∈ L1(Rn), then

v ∈ C([0,∞), L1(Rn)). (37.15)

We shall use the following lemma:

Lemma 37.6. For 1 < q ≤ 2 and u0 ∈ L∞ ∩ L1(Rn), u0 ≥ 0, the solution of(37.12) satisfies the estimate

‖v(t)‖∞ ≤ C(‖u0‖1 + ‖u0‖∞)(1 + t)−(n+1+2r)/(2q). (37.16)

Proof. Assume that a = |a|e1 without loss of generality.Step 1. Set z := vq−1 and w := zx1 = (q − 1)vq−2vx1 . We claim that

w(x, t) ≤ r + 1q|a| t−r−1, x ∈ R

n, t > 0. (37.17)

By the strong maximum principle (apply Proposition 52.7 in any bounded subdo-main) we have v > 0 in R

n × (0,∞) (unless v ≡ 0). By continuous dependence, itthus suffices to establish (37.17) when u0 also satisfies u0 ≥ ε > 0, hence v ≥ ε.The function z verifies

zt −∆z +q − 2q − 1

|∇z|2z

= −q|a|(1 + t)rzzx1.

By parabolic regularity results, w ∈ C2,1(Rn × (0,∞)). Differentiating in x1, weget

wt−∆w +2(q − 2)q − 1

∇z · ∇w

z+

2− q

q − 1|∇z|2

z2w = −q|a|(1+ t)r(w2 + zwx1). (37.18)

Since 1 < q ≤ 2, for each t0 ∈ (0, 1], the function w(t) = r+1q|a| (t + t0)−r−1 is a

supersolution of (37.18). On the other hand, for fixed τ > 0, by taking t0 smallenough, we can ensure that w(τ) < w(0). Since, for t ≥ τ , z,∇z are boundedand z is bounded away from 0 (due to (37.14) and v ≥ ε), it follows from a smallmodification of the comparison principle in Proposition 52.6 that

w(x, τ + t) ≤ w(x, t) ≤ r + 1q|a| t−r−1, x ∈ R

n, t > 0.

Claim (37.17) follows by letting τ → 0.Step 2. Write x = (x1, x

′). We claim that

‖h(t)‖∞ ≤ ‖u0‖1(4πt)−(n−1)/2, where h(x′, t) =∫ ∞

−∞v(x1, x

′, t) dx1. (37.19)

Formally, by integrating (37.12) on R with respect to x1, we see that h solvesht−∆h = 0 in R

n−1× (0,∞), so that (37.19) would follow as a consequence of theL1-L∞-estimate. However, integration needs to be justified and we shall proceedinstead as follows. For fixed R > 0, letting hR(x′, t) =

∫ R

−R v(x1, x′, t) dx1 and

integrating (37.12) on (−R, R) with respect to x1, we obtain

∂thR −∆x′hR =[(

vx1 − |a|(1 + t)rvq)(x1, x

′, t)]Rx1=−R

, x′ ∈ Rn−1, t > 0.

Fix 0 < τ < T <∞. It follows from (37.14) and (37.15) that

v(x, t), vx1 (x, t) → 0, |x| → ∞, (37.20)

uniformly for t ∈ [τ, T ]. Therefore,

∂thR −∆x′hR ≤ ε(R), x′ ∈ Rn−1, τ ≤ t ≤ T,

where limR→∞ ε(R) = 0. For x′ ∈ Rn−1 and t ∈ [τ, T ], it follows from the maxi-

mum principle that

hR(x′, t) ≤ (Gt−τ ∗ hR(τ))(x′) + ε(R)(t− τ).

By the L1-L∞-estimate, we deduce that

hR(x′, t) ≤ (4π(t− τ))−(n−1)/2‖hR(τ)‖L1(Rn−1) + ε(R)t

≤ (4π(t− τ))−(n−1)/2‖v(τ)‖L1(Rn) + ε(R)T.

Letting R →∞ and then τ → 0, using (37.15), we deduce (37.19).Step 3. By (37.20) and (37.17), we have

vq(x1, x′, t) = q/(q − 1)

∫ x1

−∞(vq−1)x1v(y1, x

′, t) dy1 ≤ Ct−r−1h(x′, t).

This, combined with (37.19), yields (37.16) for t ≥ 1, whereas (37.13) gives (37.16)for t ≤ 1.

Completion of proof of Theorem 37.4. Let U(x, t) = (1 + t)mv, where v is asolution of (37.12) for r = m(q−1) and m > 0 to be fixed later on. We shall provethat if ‖u0‖1 + ‖u0‖∞ is small enough, then U is a supersolution of (37.2) (hencev ≤ U by the comparison principle in Proposition 52.16). We have

Ut −∆U = (1 + t)m(vt −∆v) + m(1 + t)m−1v = −a · ∇(U q) + m(1 + t)m−1v.

Therefore, it will be enough to see that m(1+t)m−1v ≥ (1+t)mpvp or equivalently:

‖v(t)‖∞ ≤ m1/(p−1)(1 + t)−m−1/(p−1). (37.21)

But, since p > p1 = 1+2q/(n+1), we may choose m > 0 so small that m+1/(p−1)≤ (n + 1 + 2m(q − 1))/2q, and (37.21) follows from the lemma. The proof ofTheorem 37.4 is complete.

38. A priori bounds and blow-up rates

The following result shows that universal bounds of the form (26.25), known forthe model problem (15.1), remain valid for the perturbed problem (34.2) if theperturbation term is not too strong. In particular, this implies a (universal) a prioribound for global solutions and the usual blow-up rate estimate.

Theorem 38.1. Let p > 1 and T > 0. Assume that either

p < pB

orp < pS , Ω = R

n or BR, u = u(|x|, t), g = g(u, |ξ|).

Assume in addition that the function g : R+ × Rn → R satisfies the growth as-

sumption

|g(u, ξ)| ≤ C0(1 + |u|p1 + |ξ|q),

for some 1 ≤ p1 < p and 1 < q < 2p/(p + 1).(38.1)

38. A priori bounds and blow-up rates 339

Then, for any nonnegative classical solution of

ut −∆u = up + g(u,∇u), x ∈ Ω, 0 < t < T,

u = 0, x ∈ ∂Ω, 0 < t < T,

(38.2)

there holds

u(x, t)+ |∇u(x, t)|2/(p+1) ≤ C(1+t−1/(p−1)+(T−t)−1/(p−1)

), x ∈ Ω, 0 < t < T,

with C = C(p, p1, q, C0, Ω) > 0.

Assumption (38.1) is satisfied for instance for problems (34.3) and (34.4) whenq < 2p/(p + 1) or q < (p + 1)/2, respectively. The method of proof is based onrescaling and doubling arguments, already used in the proof of Theorem 26.8. Notethat this method does not use any variational structure, and is thus well adaptedto problem (38.2).

Proof. Since the proof is very similar to that of Theorem 26.8, we only sketchthe main changes. Instead of (26.34), we define the functions Mk by

Mk := u(p−1)/2k + |∇uk|(p−1)/(p+1).

Rescaling similarly as in (26.39) with again

λk := M−1k (xk, tk)→ 0,

the function vk is now a solution of the equation

∂svk −∆yvk = vpk + gk, (y, s) ∈ Dk,

vk = 0, y ∈ λ−1k (∂Ω− xk), |y| < k/2, |s| < k2/4,

with

gk(y, s) := λ2p/(p−1)k g

(λ−2/(p−1)k vk(y, s), λ−(p+1)/(p−1)

k ∇vk(y, s)),

v(p−1)/2k (0) + |∇vk|(p−1)/(p+1)(0) = 1,

andv(p−1)/2k + |∇vk|(p−1)/(p+1) ≤ 2, (y, s) ∈ Dk.

The growth assumption (38.1) then implies

|gk| ≤ Cλmk in Dk, where m := min

2(p− p1)p− 1

,2p− q(p + 1)

p− 1

> 0.

Now, as in the proof of Theorem 26.8, we distinguish the cases (26.42) and (26.43).By using parabolic Lp-estimates, we obtain a subsequence of vk converging to anonnegative solution v of (21.1) or (26.45). The difference is that we now use con-vergence in C1+σ,σ/2(Rn×R), which is satisfied due to the embedding (1.2). There-fore we get v(p−1)/2(0) + |∇v|(p−1)/(p+1)(0) = 1, so that v is nontrivial (moreoverv and ∇v are bounded). As before, we reach a contradiction with a Liouville-typetheorem.

Remarks 38.2. Blow-up rate. (i) For problem (34.2), the lower blow-up esti-mate ‖u(t)‖∞ ≥ C(p)(T − t)1/(p−1) is true whenever g satisfies g(u, 0) ≤ 0 and ublows up in L∞-norm (see Theorem 35.1 for a sufficient condition). This followsfrom the proof of Proposition 23.1.

(ii) By using different arguments, based on a modification of the method of theauxiliary function J of [219], the upper blow-up estimate

‖u(t)‖∞ ≤ C(T − t)−1

p−1 , 0 ≤ t < T

(this time with C depending on u) was obtained in [131] for equation (34.3) withq < 2p/(p + 1), under different assumptions. Namely, no restriction is made onp > 1, but the solution is assumed to satisfy ut ≥ 0.

In the rest of this section, we shall see that the conclusions of Theorem 38.1may become false for stronger perturbation terms in equation (34.2) (so that thegrowth restriction q < 2p/(p + 1) in (38.1) is not purely technical — although itis presently unknown whether it is optimal).

First, concerning a priori estimates of global solutions, we just recall Theo-rems 36.4 and 36.7, which already provide us with examples of global solutionsof (34.3), unbounded as t → ∞, whenever q ≥ p (in, e.g., Ω = R

n). A furthercounter-example in that direction can be found in [156] for (34.3) with p > q = 2,n = 1, Ω = (−1, 1). In that example, the solution stabilizes (monotonically) ininfinite time to a stationary solution singular at x = 0.

Next, we shall show that for stronger absorbing gradient terms, the blow-uprate may become faster, or type II [264]. Let us consider the following problem

ut − uxx = (u + 1)p − λu2

x

u + 1, − 1 < x < 1, t > 0,

u = 0, x = ±1, t > 0,

u(x, 0) = u0(x), − 1 < x < 1,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (38.3)

with p > 1 and λ ≥ 0. Note that (38.3) is of the form (34.2) (with g(u, ux) =(u + 1)p − up − λ

u2x

u+1 ).

Theorem 38.3. Consider problem (38.3) with λ > p > 1. Assume that u0 ∈ X+

is even and nonincreasing in |x|. If T := Tmax(u0) < ∞, then

(T − t)1/(p−1)‖u(t)‖∞ →∞ as t→ T. (38.4)

Remarks 38.4. (i) Instability of the blow-up rate. It was moreover provedin [264] that the assumption on λ in Theorem 38.3 is optimal: If 0 < λ ≤ p (andut ≥ 0), then the usual blow-up rate is verified:

C1 ≤ (T − t)1/(p−1)‖u(t)‖∞ ≤ C2, 0 < t < T,

for some constants C1, C2 > 0. This shows a phenomenon of strong sensitivity togradient perturbations (with λ = p being the threshold value for problem (38.3)).

(ii) It is unknown whether or not the value q = 2p/(p + 1) in Theorem 38.1 isoptimal. However, observe that the PDE in (38.3), rewritten in terms of v := u+1,has the same scale invariance properties as that in (34.3) for q = 2p/(p + 1)(cf. Remark 37.3(i)). Theorem 38.3 thus suggests that the dividing line for problem(34.3) could be given by the scaling. Namely, global unbounded solutions mightexist for q > 2p/(p + 1), and type II blow-up for 2p/(p + 1) < q < p (or even forq = 2p/(p + 1) and µ large). This conjecture is also supported by Theorem 39.1below.

Theorem 38.3 will be deduced from a result of [264] on dead-cores for theabsorption problem

wt − wxx = −wr, − 1 < x < 1, t > 0,

w(±1, t) = k, t > 0,

w(x, 0) = w0(x), − 1 < x < 1,

⎫⎪⎬⎪⎭ (38.5)

where 0 < r < 1 and k > 0. Indeed, following [304], we notice that (38.3) istransformed into (38.5) by the change of unknown

u + 1 = aw−m, m =1

λ− 1, a = m1/(p−1), (38.6)

withr =

λ− p

λ− 1∈ (0, 1), k = (λ− 1)(1−λ)/(p−1). (38.7)

Now w is nondecreasing in |x| and blow-up of u at t = T is equivalent to theappearance of a dead-core for w, i.e. w(T, 0) = 0. Note that w ≥ 0 exists for alltimes t > 0, with w ∈ C2,1([−1, 1] × (0,∞)). The fast blow-up estimate (38.4)becomes equivalent to

limt→T

(T − t)−αw(0, t) = 0, α =1

1− r. (38.8)

The proof of (38.8) relies on backward similarity variables, a tool that we havealready used in Section 25. Namely, following [244] (see also [228]) and [217], set

T − t = e−s, y = x/√

T − t and w(x, t) = (T − t)αv(y, s).

Then v satisfies the equation

vs = vyy −y

2vy + αv − vr in D, (38.9)

where D := (y, s) : − log T < s < ∞, |y| < es/2. Under the assumptions ofTheorem 38.3, we shall actually show the more precise convergence statement

lims→∞ v(y, s) = V1(y) := kr|y|2α, kr =

[ (1− r)2

2(1 + r)

]α, (38.10)

uniformly on |y| < R for each R > 0, from which (38.8) (and hence (38.4))readily follows.

A quick check reveals that the right-hand side V1(y) of (38.10) provides an(unbounded) stationary solution of (38.9), more precisely a solution of

Vyy −y

2Vy + αV − V r = 0, y ∈ R. (38.11)

Note that each solution of (38.11) corresponds to a self-similar solution of wt =wxx − wr in R × (−∞, T ) given by w(x, t) = (T − t)1/(1−r)V (x/

√T − t). On the

other hand V1(x), restricted to [−1, 1], is also a stationary solution of (38.5) withk = kr.

The proof of Theorem 38.3 will then be carried out in three steps.

(i) Identify the stationary solutions of (38.9) (in a suitable set);(ii) Prove that all the global solutions of (38.9) are attracted by the set of

stationary solutions of (38.9) (in the locally compact topology);(iii) Discard all the possible limits other than the stationary solution V1.

We need three lemmas. In what follows, we shall use the fact that, by parabolicregularity results, wx ∈ C2,1((−1, 1)× (0, T )) ∩ C([−1, 1]× (0, T ]). We start witha lower estimate which is the key ingredient in step (iii).

Lemma 38.5. Let u satisfy the hypotheses of Theorem 38.3, and w be defined by(38.6)–(38.7). There exists c1 > 0 (depending on u) such that

w(x, t) ≥[w1−r(0, t) + c1x

2]α

, |x| ≤ 1, T/2 ≤ t ≤ T. (38.12)

Proof. The basic idea of the proof is similar to that in Theorem 24.1 (cf. [219]),but some special care is required and an auxiliary nonlocal parabolic equation hasto be considered — cf. (38.17) below. We set

J = wx − εxwr .

It will be sufficient to show that, for ε > 0 small enough, there holds

J ≥ 0 in [0, 1]× (T/2, T ). (38.13)

Indeed, we will then have (w1−r)x = (1− r)w−rwx ≥ ε(1− r)x in [0, 1]× (T/2, T )and the estimate will immediately follow by integrating in space between 0 and x.


To prove (38.13), we first claim that for ε > 0 sufficiently small, we have

J(x, T/2) > 0 in (0, 1], Jx(0, t) > 0 and J(1, t) > 0 on (T/2, T ). (38.14)

We have wx ≥ 0 in [0, 1]×(0, T ] and wx(0, t) = 0 in (0, T ]. Next, since w(x, t) ≤ k(due to w0 ≤ k), we have wx(1, t) > 0 in (0, T ] by Hopf’s lemma (cf. Proposi-tion 52.7). Moreover, since z := wx satisfies zt − zxx = −rwr−1z in [0, 1]× (0, T ),the strong maximum principle (Proposition 52.7) then implies

wx(x, t) > 0 in (0, 1]× (0, T ]. (38.15)

As z achieves its minimum value z = 0 at x = 0 for each t ∈ (0, T ), we also have

wxx(0, t) = zx(0, t) > 0 in (0, T ), (38.16)

in view of Hopf’s lemma. The claim (38.14) follows from (38.15) and (38.16).Let us now compute

(xwr)t = xrwr−1wt,

(xwr)x = wr + xrwr−1wx,

(xwr)xx = 2rwr−1wx + xrwr−1wxx + xr(r − 1)wr−2(wx)2.

We get

Jt − Jxx = (wt − wxx)x − ε(xwr)t + ε(xwr)xx

= −rwr−1wx + ε(−xrwr−1wt + 2rwr−1wx

+ xrwr−1wxx + xr(r − 1)wr−2(wx)2)

= −rwr−1(wx + εx(wt − wxx)) + εrwr−2wx(2w + x(r − 1)wx))

= −rwr−1J + εrwr−2wx

(2w + x(r − 1)J + εx2(r − 1)wr)

).

Putting

a(x, t) = rwr−1 + εxr(1 − r)wr−2wx and b(x, t) = 2εr(1− r)w2r−2wx,

we obtainJt − Jxx + aJ = εrwr−2wx(2w − εx2(1 − r)wr)

= b(w1−r

1− r− ε

x2

2

).

On the other hand, we note that

(w1−r

1− r− ε

x2

2

)x= wxw−r − εx = w−rJ.

It follows that

Jt − Jxx + aJ = b

(w1−r(0, t)

1− r+∫ x

0

w−rJ(y, t) dy

). (38.17)

We can then apply a simple nonlocal version of the maximum principle to deducethat J ≥ 0. The key point which allows this is that the function b is positive. Letus give the details to make everything safe.

By continuity, using (38.14), we have

E :=τ ∈ (T/2, T ) : J > 0 on (0, 1]× [T/2, τ)

= ∅.

Assume for contradiction that t0 := sup E < T . Then there holds J(x, t0) ≥ 0in [0, 1] and there exists x0 ∈ (0, 1) such that J(x0, t0) = 0, Jt(x0, t0) ≤ 0 andJxx(x0, t0) ≥ 0. Substituting this into (38.17) and noting that b(x0, t0) > 0, weobtain

0 ≥(Jt − Jxx + aJ

)(x0, t0) = b(x0, t0)

(w1−r(0, t0)

1− r+∫ x0

0

w−rJ(y, t0) dy

)> 0.

This contradiction shows that t0 = T , which gives the desired conclusion.

Lemma 38.6. Let u satisfy the hypotheses of Theorem 38.3, and w be defined by(38.6)–(38.7). There exists c2 > 0 (depending on u) such that

w(x, t) ≤[w(1−r)/2(0, t) + c2|x|

]2α (38.18)

for all T/2 ≤ t ≤ T , |x| ≤ 1. Moreover, the corresponding global solution v of(38.9) satisfies

v(y, s) ≤ C(1 + |y|)2α and |vy(y, s)| ≤ C(1 + |y|)2α−1 (38.19)

for all − log(T/2) =: s0 < s < ∞, |y| < es/2.

Proof. We consider the function J(x, t) := 12w2

x − Cwr+1, where C > 1 is aconstant to be determined later. We compute

Jt − Jxx = wx(wt − wxx)x − w2xx − C(r + 1)

[wr(wt − wxx)− rwr−1w2

x

]= −rwr−1w2

x − w2xx + C(r + 1)

[w2r + rwr−1w2

x

]= C(r + 1)w2r + r(C(r + 1)− 1)wr−1w2

x − w2xx

in (0, 1)× (T/2, T ). Using the relation w2x = 2(J + Cwr+1), we get

Jt − Jxx + b1J = C[1− r + 2r(r + 1)C]w2r − w2xx,

with b1(x, t) = 2(1−C(r+1))rwr−1. Then using wxx = Jx/wx+C(r+1)wr (recallthat wx > 0 in (0, 1)× (T/2, T )) and setting b2(x, t) = Jx/w2

x + 2C(r + 1)wr/wx,we end up with

Jt − Jxx + b2Jx + b1J = C(1− r)[1 − (r + 1)C]w2r < 0.

Now, for C > 0 sufficiently large, we have J < 0 on the parabolic boundary ofQ := (0, 1)× (T/2, T ). The maximum principle then yields J ≤ 0 in Q, hence

(w(1−r)/2)x =1− r

2wxw−(r+1)/2 ≤ C′

and the estimate (38.18) follows. Note that we get in turn the estimate

|wx| ≤ Cw(r+1)/2, |x| ≤ 1, T/2 ≤ t ≤ T. (38.20)

Let us next prove (38.19). Since wxx(0, t) ≥ 0, we have wt(0, t) ≥ −wr(0, t). Byintegrating between t and T , we easily get w(0, t) ≤ C(T − t)α. By combining thiswith (38.18), we obtain

v(y, s) = (T − t)−αw(y√

T − t, t) ≤ (T − t)−α[w(1−r)/2(0, t) + c2|y|

√T − t

]2α

≤ C(T − t)−α[√

T − t + |y|√

T − t]2α

= C(1 + |y|)2α.

The estimate of vy then follows from (38.20). The proof of the lemma is com-plete.

Next, for step (i), we have the following.

Lemma 38.7. Let V ∈ C2(R) be a solution of (38.11) such that

V = V (|y|), with V ′ ≥ 0, V > 0 for all y > 0,

and such that V is polynomially bounded. Then

V = V1 := kr|y|2/(1−r) or V = V2 := κ := (1 − r)1/(1−r).

Proof. Let W := V 1−r and denote ′ = d/dy. Since V > 0 for y > 0, W is smooththere. The equation for W is:

W ′′ − y

2W ′ +

r

1− r

W ′2

W+ W = 1− r. (38.21)

By differentiating, we note that

W ′′′ − y

2W ′′ +

12W ′ = − r

1− r

(W ′2

W

)′= − r

1− r

W ′(2W ′′W −W ′2)W 2

. (38.22)

Set H := W − y2W ′ and let D = y > 0 : H(y) = 0. For all y ∈ D, Z := |H | 1

1−r

is smooth and we compute

Z ′ =1

1− r|H |

2r−11−r HH ′, Z ′′ =

11− r

|H |2r−11−r

(HH ′′ +

r

1− rH ′2

),

hencey

2Z ′ − Z ′′ =

11− r

|H |2r−11−r

(H(y

2H ′ −H ′′

)− r

1− rH ′2

),

andH ′ =

12W ′ − y

2W ′′, H ′′ = −y

2W ′′′.

Using (38.22), it follows that, for all y ∈ D,

y

2Z ′ − Z ′′

=1

1− r|H |

2r−11−r

y

2

(W − y

2W ′)(

W ′′′ − y

2W ′′ +

W ′

2

)− r

1− r

(W ′

2− y

2W ′′)2

= − r

4(1− r)2|H |

2r−11−r

y(2W − yW ′)

W ′(2W ′′W −W ′2)W 2

+ (W ′ − yW ′′)2

= − r

4(1− r)2|H |

2r−11−r

W ′2 + y2

(WW ′′ −W ′2

W

)2

+ 2yW ′(WW ′′ −W ′2

W

)= − r

4(1− r)2|H |

2r−11−r

(W ′ + y

(WW ′′ −W ′2

W

))2

≤ 0

hence(e−y2/4Z ′)′ ≥ 0 in D. (38.23)

We next claim that the function Z ≥ 0 is nonincreasing in (0,∞). Indeed,otherwise, there would exist y0 such that Z(y0) > 0 and Z ′(y0) > 0, hence Z ′ ≥Cey2/4 for y ≥ y0 by (38.23). Due to |(y−2W )′| = 2y−3Z1−r, we would get W ≥eηy2

as y → ∞, for some η > 0, contradicting the polynomial bound assumed onV .

Now assume for contradiction that Z is nonconstant on (0,∞). Then there isR > 0 such that Z(0) > Z(R) and we may choose ε > 0 so small that f :=Z + εey2/2 satisfies f(0) > Z(0) > f(R). It follows that f has a local maximum atsome y1 ∈ (−R, R) and that Z(y1) > 0 (hence y1 ∈ D). Therefore, at y = y1, weget 0 ≤ (y/2)f ′ − f ′′ ≤ ε((y2/2)− 1− y2)ey2/2 < 0, a contradiction.

We deduce that W − (y/2)W ′ = C on (0,∞). By integration, we finally getW = A + By2 and the conclusion follows easily by substituting into equation(38.21).

Now, the rest of the proof of Theorem 38.3 via the dead-core rate estimate(38.8), and in particular Step (ii), will be a consequence of energy arguments closeto those from Section 25 (cf. [244]) for blow-up problems. A difference with [244],[217] is that here v is not uniformly bounded; and indeed it will be proved that,unlike in those works, v converges to an unbounded self-similar profile.

Proof of Theorem 38.3. Let ρ(y) = e−y2/4 and define R(s) = es/2 and

E(s) =∫ R(s)

0

(v2y

2+

vr+1

r + 1− αv2

2

)(y, s)ρ(y) dy.

For s ≥ s0 = − log(T/2), we have

E′(s) = R′(s)ρ(v2

y

2+

vr+1

r + 1− αv2

2

)(R(s), s) +

∫ R(s)

0

(vyvys + (vr − αv)vs

)ρ dy

= ρ

[R′(s)

(v2y

2+

vr+1

r + 1− αv2

2

)+ vyvs

](R(s), s)

+∫ R(s)

0

(−(ρvy)y + ρ(vr − αv)

)vs dy

= ρ

[R′(s)

(v2y

2+

vr+1

r + 1− αv2

2

)+ vyvs

](R(s), s)−

∫ R(s)

0

ρv2s dy

≡ A(s)−∫ R(s)

0

ρv2s dy.

On the other hand, by (38.19), we have

|E(s)| ≤∫ ∞

0

Ce−y2/4(1 + |y|)4α dy = C, s ≥ s0

and, using vs(R(s), s) =(αv − y

2vy

)(R(s), s) and (38.19), we obtain

|A(s)| ≤ C exp(−1

4es)es/2(1 + es/2)4α,

hence A(s) ∈ L1(s0,∞). It follows that

∫ ∞

s0

∫ R(s)

0

ρv2s dy ds < ∞.

Then, by arguing similarly as in the proof of Lemma 25.6(i), we deduce that, foreach sequence sn →∞, there exists a subsequence s′n such that v(·, s′n) convergesto a solution V of (38.11), uniformly on |y| < R for each R > 0.

But on the other hand, by the lower bound (38.12), for each y ∈ R and s >2 log |y|, we have

v(y, s) = (T − t)−αw(y√

T − t, t)≥ (T − t)−α

(c1|y

√T − t|2

)α= cα1 |y|2α.

In view of Lemma 38.7 and (38.19), this shows that necessarily V = V1. Theconclusion follows.

39. Blow-up sets and profiles

The following results show that there is a threshold q = 2p/(p + 1) above whichthe absorbing gradient term has a strong influence on the final blow-up profile ofsolutions of (34.3), making it more and more singular as q increases to p (observethat q/(p − q) > 2/(p− 1) for 2p/(p + 1) < q < p). Theorems 39.1 and 39.2 arefrom [131] and [490], respectively.

Theorem 39.1. Consider problem (34.3) with 1 < q < p, µ > 0, and Ω = BR.Let u0 ∈ X+ be radial nonincreasing and such that T := Tmax(u0) < ∞. Then 0is the only blow-up point. Moreover, for all α > α0, there holds

u(r, t) ≤ Cαr−α, 0 ≤ t < T, 0 < r ≤ R,

with

α0 =

2/(p− 1) if 1 < q ≤ 2p/(p + 1),q/(p− q) if 2p/(p + 1) < q < p.

The optimality of Theorem 39.1 is shown by the following:

Theorem 39.2. Under the hypotheses of Theorem 39.1, assume in addition thatut ≥ 0 in QT . Then there exist C, η > 0 such that

u(r, T ) := limt→T

u(r, t) ≥ Cr−α0 , 0 < r < η. (39.1)

Remarks 39.3. (i) The assumption ut ≥ 0 is guaranteed if u0 is a subsolution ofthe stationary problem (see Proposition 52.19), and it is not difficult to constructsuch initial data.

(ii) We have η = η(u0) > 0 in (39.1), but we may take C = C(p) > 0 if1 < q ≤ 2p/(p + 1), C = C(p, q, µ) > 0 if 2p/(p + 1) < q < p.

The proof of Theorem 39.1 consists of two steps. The first one (Lemma 39.4)is a modification of the argument of [219] (cf. the proof of Theorem 24.1), whichconsists in estimating −ur from below near r = 0, by applying the maximumprinciple to an auxiliary function of the form J = ur + cε(r)F (u). The second step(in the case q > 2p/(p + 1)) is an additional bootstrap argument (on the value ofγ for F (u) = uγ), which enables one to reach the optimal exponent α0.

39. Blow-up sets and profiles 349

Lemma 39.4. Consider problem (34.3) with 1 < q < p, µ > 0, Ω = BR, and letu0 be as in Theorem 39.1. Denote f(u) = up and let F ∈ C2((0,∞)) ∩C1([0,∞))satisfy F, F ′, F ′′ ≥ 0, with F ′′F bounded near 0. Let δ > 0 and set K = (q − 1)µ.Assume that

G(y) :=∫ ∞

y

ds

F (s)< ∞, y > 0

and that, for all sufficiently small ε > 0,

f ′F − fF ′ + ε2r2+2δF ′′F 2 + δ(n + δ)r−2F

≥ 2ε(1 + δ)rδF ′F + 2q−1Kεqrq+qδF qF ′, u > 0, 0 < r ≤ R.(39.2)

Then 0 is the only blow-up point and there exists ε0 > 0 such that

u(r, t) ≤ G−1(ε0r2+δ) in [0, R]× [T/2, T ).

Remark 39.5. If u is a given solution satisfying the assumptions of Lemma 39.4,then the conclusion remains valid if (39.2) is assumed to hold for all r and allu = u(r, t) such that (r, t) ∈ (0, R) × [T/2, T ) (instead of for all u > 0 and0 < r < R). This fact will be used in the proof of Theorem 39.1(ii).

Proof of Lemma 39.4. Set

J = w + cε(r)F (u)

where cε(r) = εr1+δ and w = ur ≤ 0. By parabolic regularity results, we havew ∈ C2,1((0, R)× (0, T )) ∩ C([0, R]× (0, T )). Also, u > 0 in [0, R)× (0, T ) by thestrong maximum principle. Differentiating the equation


rur = f(u)− µ|ur|q

with respect to r yields

wt − wrr −n− 1

rwr = −n− 1

r2w + f ′(u)w + qµ|w|q−1wr. (39.3)

Using (39.3) and writing f, F for f(u), F (u), we compute the equation for J :

Jt − Jrr −n− 1

rJr = −n− 1

r2w + f ′(u)w + qµ|w|q−1wr + cε[f − µ|w|q ]F ′

− 2wc′εF′ −[n− 1

rc′ε + c′′ε

]F − cεw

2F ′′.

Using the relations w = J − cεF , w2 = c2εF

2 + (J − 2cεF )J and

wr = Jr − c′εF − cεF′w,

we obtain

Jt − Jrr −(n− 1

r+ qµ|w|q−1

)Jr − b0J

= cε

(−f ′ − c′ε

cεqµ|w|q−1 +

n− 1r2

− c′′εcε− n− 1

r

c′εcε

)F

+ (f + K|w|q)F ′ + 2c′εF′F − c2

εF′′F 2

where

b0 = f ′ − (n− 1)r−2 − 2c′εF′ − cε(J − 2cεF )F ′′.

The assumption (39.2) is equivalent to

f ′F − fF ′ − 2c′εF′F + c2

εF′′F 2 − 2q−1Kcq

εFqF ′ +

(c′′εcε

+n− 1

r

c′εcε− n− 1

r2

)F ≥ 0.

Combining this with |w|q ≤ 2q−1(|J |q + cqεF

q), we obtain

Jt − Jrr −(n− 1

r+ qµ|w|q−1

)Jr − bJ ≤ 0 in (0, R)× (0, T ],

where b = b0 + 2q−1KcεF′|J |q−2J .

On the other hand, arguing as in the proof of Theorem 24.1, we obtain ur < 0in (0, R]× (0, T ) and urr(0, t) < 0 in (0, T ). It follows that J(·, T/2) ≤ 0 in [0, R]for ε small. Obviously J(0, t) ≤ 0 and J(R, t) < 0 for all t ∈ (0, T ). Since bis bounded above in ((0, R) × (0, τ)) ∩ (x, t) : J > 0 for each τ ∈ (0, T ), itfollows from the maximum principle (cf. Proposition 52.4 and Remark 52.11(a))that J ≤ 0 in [0, R]× [T/2, T ). Integrating this inequality between 0 and r yieldsthe conclusion.

We shall show that condition (39.2) in Lemma 39.4 is satisfied for F (u) = uγ

with suitable choices of γ > 1. The inequality (39.2) takes the form

(p− γ)up+γ−1 + (εr1+δ)2γ(γ − 1)u3γ−2 + δ(n + δ)r−2uγ

≥ 2εγ(1 + δ)rδu2γ−1 + 2q−1Kγ(εr1+δ)quγq+γ−1.

In the proof of this inequality, we shall need the following elementary lemma.

Lemma 39.6. Let n be a positive integer, R, K, δ > 0 and p > 1.(i) If 1 < γ < p, then for ε > 0 small enough, there holds

12

((p− γ)up+γ−1 + δ(n + δ)r−2uγ

)≥ 2εγ(1 + δ)rδu2γ−1, 0 < r ≤ R, u ≥ 0.

(39.4)

(ii) If 1 < q < 2p/(p + 1) and γ ∈ (p/q, p), then for ε > 0 small enough, thereholds

12

((p− γ)up+γ−1 + (εr1+δ)2γ(γ − 1)u3γ−2

)≥ 2q−1Kγ(εr1+δ)quγq+γ−1, 0 < r ≤ R, u ≥ 0.

(39.5)

(iii) If 1 < q 0 small enough, there holds

12(p− γ)up+γ−1 ≥ 2q−1Kγ(εr1+δ)quγq+γ−1, 0 < r ≤ R, u ≥ 0. (39.6)

Proof. Inequalities (39.4) and (39.5) are consequences of Young’s inequality

aα

α+

bβ

β≥ ab, a, b ≥ 0, α, β > 1,

1α

+1β

= 1,

where we choose α = (p − 1)/(γ − 1) in the case of (39.4) and α = (2γ − 1 − p)/(2γ − 1 − γq) in the case of (39.5). In inequality (39.6), it is sufficient to chooseε ≤ ( p−γ

2qKγ )1/qR−1−δ.

We now continue with the proof of Theorem 39.1.

Proof of Theorem 39.1. First assume 1 < q < 2p/(p + 1). In this case, wechoose F (u) = uγ with 1 < γ < p and Lemma 39.6(i) and (ii) yields that (39.2)holds. Lemma 39.4 then implies

u(r, t) ≤ Cr−2+δγ−1

and 2+δγ−1 can be made arbitrarily close to 2/(p− 1).

Next consider the case 2p/(p+ 1) ≤ q < p. Now we first choose F (u) = uγ withγ = p/q and Lemma 39.6(i) and (iii) yields that (39.2) holds. Lemma 39.4 implies

u(r, t) ≤ Cr−α, α = α(δ, γ) =2 + δ

γ − 1. (39.7)

Inequality (39.6) is equivalent to

uγq−p ≤ p− γ

Kγ(2εr1+δ)−q

and, due to the estimate (39.7) on u, it is also true (for u = u(r, t) — cf. Re-mark 39.5) if γ is replaced by γ ∈ (γ, p) such that (γq − p)α < (1 + δ)q, or,equivalently,

γ <p

q+

1 + δ

2 + δ(γ − 1).

If δ is chosen small enough, this reduces to

γ <p

q+

γ − 12

.

Clearly,

γ <p

q+

γ − 12

if γ <2p

q− 1 (≤ p),

andα(δ, γ)→ q

p− qas δ → 0, γ → 2p

q− 1.

Consequently, an obvious bootstrap argument implies the assertion.

Proof of Theorem 39.2. We modify the argument in the proof of Theorem 24.3.Step 1. We claim that

‖ur(t)‖∞ ≤ C1uγ(0, t), (39.8)

with γ = min((p + 1)/2, p/q

)> 1.

On the one hand, since ut ≥ 0 and ur ≤ 0, we have

∂

∂r

(12u2

r +1

p + 1up+1

)= (urr + up)ur =

(ut + µ|ur|q −

n− 1r

ur

)ur ≤ 0,

hence (12u2

r +1

p + 1up+1

)(r, t) ≤ 1

p + 1up+1(0, t).

Therefore, we get (39.8) with γ = (p + 1)/2 (and C1 = C1(p)).On the other hand, for each t ∈ (0, T ), at a point r ∈ (0, R] where |ur(·, t)|

achieves its maximum, we have

µ|ur|q = up + urr − ut +n− 1

rur ≤ up,

due to ut ≥ 0, ur ≤ 0 and urr(r, t) ≤ 0. This yields (39.8) with γ = p/q (andC1 = µ−1/q), hence the claim.

Step 2. For 0 < t < T := Tmax(u0), let r0(t) be such that u(r0(t), t) = 12u(0, t).

Note that, since ur < 0 for 0 < t < T and 0 < r ≤ R, the implicit functiontheorem guarantees that r0(t) is unique and is a continuous function of t. SinceTmax(u0) < ∞, we have u(0, t) → ∞ as t → T , due to Theorem 35.1. Also, byTheorem 39.1, we know that 0 is the only blow-up point, hence r0(t) → 0 as t→ T .Now we have

−ur ≤ C2uγ , 0 ≤ r ≤ r0(t).

Integrating, we get

u1−γ(r0(t), t) ≤ u1−γ(0, t) + C3r0(t) = 21−γu1−γ(r0(t), t) + C3r0(t),

hence u(r0(t), t) ≥ C4(r0(t))−1/(γ−1). Using ut ≥ 0, it follows that

u(r0(t), T ) ≥ C4(r0(t))−1/(γ−1), 0 < t < T.

Since r0 is continuous and r0(t)→ 0 as t→ T , we deduce that the range r0((0, T ))contains an interval of the form (0, η) and the conclusion follows.

For equation (38.3), the arguments in the proof of Theorem 38.3 provide preciseinformation on the blow-up profile, which turns out to be slightly less singular thanfor the model problem (15.1) (cf. Remark 25.8).

Theorem 39.7. Consider problem (38.3) with λ > p. Assume that u0 ∈ X+

is even and nonincreasing in |x|. If T := Tmax(u0) < ∞, then for each x = 0,u(x, T ) := limt→T u(x, t) exists and it satisfies

C1 ≤ |x|2/(p−1)u(x, T ) ≤ C2, x small, x = 0,

for some constants C1, C2 > 0.

Proof. The (globally defined) solution w ≥ 0 of the transformed problem (38.5)(cf. formulas (38.6)–(38.7)) satisfies (38.12) and (38.18). In particular, by (38.12),parabolic estimates and standard embeddings, we have u ∈ BUCα(ε < |x| < 1×(T/2, T )) for each ε > 0 and some α ∈ (0, 1). It follows that the limit u(x, T ) existsfor x ∈ [−1, 1] \ 0. On the other hand, since w(0, T ) = 0, (38.12) and (38.18)imply

(c1|x|2)α ≤ w(x, T ) ≤ (c2|x|)2α, −1 < x < 1.

The assertion concerning u(x, T ) follows immediately.

Remarks 39.8. (i) Self-similar blowing-up solutions. As mentioned in Re-mark 37.3(i) when q = 2p/(p+1), the equation (34.3) is scale-invariant. This prop-erty has been exploited in [495] to construct backward self-similar (blow-up) solu-tions by ODE methods. More precisely, for each 0 < µ < 2 and 1 < p < p0(n, µ),there exists a solution of (34.3) of the form

u(x, t) = (T − t)−1/(p−1)W(x/(T − t)1/2

), (39.9)

for (x, t) ∈ Rn× (−∞, T ). Here W is a positive, C2, radially symmetric decreasing

function on Rn. Moreover, for all such solutions, W satisfies

lim|x|→∞

|x|2/(p−1)W (x) = C > 0.

This guarantees that u blows up at the single point x = 0 and admits a limitingprofile, similar to that obtained for equation (38.3) in Theorem 39.7, given by

u(x, T ) = C|x|−2/(p−1), for all x = 0.

In contrast, recall that no nontrivial, backward, self-similar solutions exist forµ = 0 and p ≤ pS (cf. Proposition 25.4).

Comparison of this result with Theorems 39.1–39.2 yields the interesting anda bit surprising observation that the gradient term can have opposite effects onthe blow-up profile: When the perturbation is mild (q = 2p/(p + 1)), the profile isslightly less singular; when the perturbation is strong (2p/(p + 1) < q < p), it ismore singular.

(ii) Single-point vs. regional blow-up. We have seen several examples ofsingle-point blow-up for equations with dissipative gradient terms in the radial case(cf. Theorems 39.1 and 39.7 and Remark 39.8(i)). Also, examples of single-pointblow-up for the convective problem (34.4) can be found in [218]. On the other hand,it was proved in [131] that if Ω is convex bounded, µ > 0 and q < 2p/(p+1), thenthe blow-up set of any solution of (34.3) is a compact subset of Ω. The situationis quite different when µ < 0. Namely, for q = 2 one has single-point blow-up ifp > 2, regional blow-up if p = 2, and global blow-up if 1 < p < 2 (see [313], [304],[231]). The proof relies on the transformation v = eu − 1, which converts (34.3)into the equation with mildly superlinear source vt −∆v = (1 + v) logp(1 + v).

The authors of [304] interpret this result in the following way. While the termup alone would force the solution to develop a spike at the maximum point, hencecausing single-point blow-up, the gradient term now has a positive sign and tendsto push up the steeper parts of the graph of u(., t). This enhances regional or evenglobal blow-up, the influence of the gradient term becoming more important asthe value of p decreases.

(iii) L∞ boundary blow-up for a Dirichlet problem. For the convectiveproblem (34.4), a surprising example was recently constructed in [202], of a solutionblowing up (only) at the boundary, in spite of the imposed homogeneous Dirichletboundary condition. More precisely, consider problem (34.4) with n = 1, Ω =(0,∞), p > 1 and q = (p + 1)/2. Then, for −a > 0 sufficiently large, there exists apositive solution u such that

limt→T

supx>0

u(x, t) = ∞

andu(x, t) ≤ C|x|−2/(p−1), x > 0, 0 < t < T.

This solution is constructed in the backward self-similar form (39.9), now withW (y) > 0, y > 0, and W (0) = 0 (note that (34.4) is scale-invariant for q =(p + 1)/2, similarly as (34.3) for q = 2p/(p + 1) — cf. Remark 39.8(i)).

(iv) More self-similar profiles. Concerning self-similar profiles, still in thecase µ < 0, q = 2, with Ω = R

n, it is proved in [231] that radial blow-up solutions toequation (34.3) behave asymptotically like a self-similar solution w of the followingHamilton-Jacobi equation without diffusion:

wt = |∇w|2 + wp.

40. Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary 355

The function w is of the form

w(x, t) = (T − t)−1/(p−1)W(x/(T − t)m

), m = (2 − p)/2(p− 1).

Note that this kind of self-similar behavior is quite different from that in (i) above(or from those known for µ = 0 and p supercritical); indeed, m describes the range(−∞, 1/2) for p ∈ (1,∞).

40. Viscous Hamilton-Jacobi equations andgradient blow-up on the boundary

In this section we study problem (34.5), which exhibits quite different phenomenafrom the model problem (15.1) or its perturbations (34.3), (34.4). For simplicitywe shall again only consider nonnegative solutions (this assumption is essential insome, but not all, of the results).

40.1. Gradient blow-up and global existence

A basic fact about (34.5) is that solutions are uniformly bounded. Indeed, as adirect consequence of the maximum principle, for any u0 ∈ X+, there holds

0 ≤ u(x, t) ≤ maxx∈Ω

u0(x), x ∈ Ω, 0 ≤ t < Tmax(u0). (40.1)

In view of (40.1), and since (34.5) is well-posed in the space X , a solution cancease to exist in finite time Tmax(u0) <∞ only if

limt→Tmax(u0)

‖∇u(t)‖∞ =∞. (40.2)

This is what we call gradient blow-up (GBU for short).

Unlike the model problem (15.1), for which nonglobal solutions exist if andonly if p > 1, the following two results show that the dividing line for the Dirichletproblem associated with (34.5) is given by p = 2.

Theorem 40.1. Consider problem (34.5) with 1 < p ≤ 2. Then Tmax(u0) = ∞for any u0 ∈ X+. Moreover, we have

supt≥0

‖u(t)‖X <∞.

Theorem 40.2. Consider problem (34.5) with p > 2 and Ω bounded, and let1 ≤ q <∞. There exists C = C(p, q, Ω) > 0 such that, if u0 ∈ X+ and ‖u0‖q ≥ C,then Tmax(u0) < ∞.

Theorem 40.1 is an immediate consequence of the boundary gradient estimatefrom Lemma 35.4 and of the following simple result, which asserts that for problem(34.5), |∇u| achieves its maximal values on the parabolic boundary.

Proposition 40.3. Assume p > 1 and u0 ∈ X+. Let u be the solution of (34.5)and let 0 < T < Tmax(u0). Then

supt∈[0,T ]

‖∇u(t)‖∞ = supPT

|∇u|.

Proof. Fix h ∈ Rn, with |h| = 1, and put w := ∂hv = h · ∇v. We have w ∈

C(QT ) ∩ L∞(QT ), and parabolic regularity results imply w ∈ C2,1(QT ). Takingthe space derivative of the equation in the direction h, we obtain

wt −∆w = b(x, t) · ∇w in QT ,

where b(x, t) = p|∇u|p−2∇u. By the maximum principle, we deduce that supQTw

≤ supPTw. Since h is arbitrary, the conclusion follows.

Proof of Theorem 40.2. Put q0 := 2(p − 1)/(p − 2). It is obviously sufficientto show the assertion for q0 ≤ q < ∞. Let thus set k := q − 1 ∈ [p/(p − 2),∞).Multiplying (34.5) by uk, we get

1k + 1

d

dt

∫Ω

uk+1(t) dx =∫

Ω

|∇u|p uk dx − k

∫Ω

|∇u|2uk−1 dx. (40.3)

On the other hand, by Poincare’s inequality, we have∫Ω

|∇u|p uk dx = C

∫Ω

∣∣∇u(p+k)/p∣∣p ≥ C

∫Ω

up+k dx. (40.4)

Since k ≥ p/(p− 2), by using Holder’s inequality and (40.4), we get∫Ω

|∇u|2uk−1 dx ≤(∫

Ω

|∇u|puk dx)2/p(∫

Ω

uk−p/(p−2) dx)(p−2)/p

≤ C(∫

Ω

|∇u|puk dx)(k+1)/(k+p)

.

Combining this with (40.3), (40.4) and Holder’s inequality, we obtain

d

dt

∫Ω

uk+1 dx ≥∫

Ω

|∇u|p uk dx− C ≥ C1

(∫Ω

uk+1 dx)(k+p)/(k+1)

− C2.

The conclusion follows.


Remarks 40.4. (i) Different methods of proof. Theorem 40.2, whose proofrelies on multiplication by powers of u, is due to [279] for q = 2(p − 1)/(p − 2)and [500] in the general case. By a different argument, using the first eigenfunc-tion, GBU for problem (34.5) was shown in [485] under a stronger condition onu0 (see also [6]). The first example of GBU seems to be due to [210], where aone-dimensional problem with time-dependent Dirichlet boundary conditions wasconsidered. The proof was based on subsolution arguments.

(ii) Sharp condition for GBU. A more precise growth condition for prevent-ing GBU is known to be

|F (u,∇u)| ≤ C(u)(1 + |∇u|2)h(|∇u|) (40.5)

where h is positive nondecreasing and satisfies∫ ∞ ds

sh(s)= ∞, (40.6)

and C(u) is locally bounded (compare with condition (17.4) in the case of L∞-blow-up); see [320], [337], [512]. There are known examples showing that condition(40.5)–(40.6) is sharp. A GBU result for general (including homogeneous) Dirichletdata can be found in [485]. The proof relies on eigenfunction and convex conju-gate functions arguments. Earlier examples involving particular time-dependentboundary data, and relying on subsolution methods, were given in [337].

Unlike in the Dirichlet problem, global existence for the Cauchy problem holdsfor any p > 1 (cf. [402], [26]):

Proposition 40.5. Consider problem (34.5) with Ω = Rn and p > 1. Then

Tmax(u0) =∞ for any u0 ∈ X+. Moreover, we have

supt≥0

‖∇u(t)‖∞ = ‖∇u0‖∞. (40.7)

Proposition 40.5 is an immediate consequence of Proposition 40.3.

Remark 40.6. Unbounded domains. Although Theorem 40.2 is stated forbounded domains, GBU for large data when p > 2 occurs in any (regular) un-bounded domain Ω other than R

n. (Thus, for p > 2, Proposition 40.5 is true onlyin R

n.)Indeed, this follows from a simple comparison argument: Choose a ball B ⊂ Ω

such that ∂B ∩ ∂Ω consists of a single point, say x0. Without loss of generality,we may assume that B = B(0, ρ) and x0 = ρ e1. Let 0 ≤ φ ∈ C1(B) satisfy φ = 0on ∂B, φ radially symmetric, and let v be the solution of problem (34.5) with Ωreplaced by B and initial data φ. If ‖φ‖1 is sufficiently large, then v has GBU in afinite time T , due to Theorem 40.2. Since v is radially symmetric, it follows from

Proposition 40.3 that lim inft→T∂v∂x1

(x0, t) = −∞. Take any u0 ∈ X+ such thatu0 ≥ φ in B and let u be the solution of (34.5). By the comparison principle, wehave u(x, t) ≥ v(x, t) in B as long as u exists. Since u(x0, t) = v(x0, t) = 0, thisimplies ∂u

∂x1(x0, t) ≤ ∂v

∂x1(x0, t). Consequently GBU must occur for u no later than

at time T .

40.2. Asymptotic behavior of global solutions

We start with the case of bounded domains. We have the following result onexponential decay. Assertions (i), (ii) follow from [146], [69], and (iii) from [485].

Theorem 40.7. Consider problem (34.5) with p > 1, Ω bounded and u0 ∈ X+.(i) Assume that

Tmax(u0) = ∞ and supt≥0

‖u(t)‖X < ∞. (40.8)

Then there exists C > 0 (depending on u), such that

‖u(t)‖X ≤ Ce−λ1t, t ≥ 0. (40.9)

(ii) If 1 2, then assertions (40.8) and (40.9) are true whenever ‖u0‖X is suffi-ciently small.

In the proof we shall use the following simple observation about steady statesof (34.5) (cf. [341]):

Proposition 40.8. Assume Ω bounded and let p > 1. Then the only solutionv ∈ C2 ∩ C0(Ω) of ∆v + |∇v|p = 0 is the trivial solution v ≡ 0.

Proof. For ε > 0 small, let us denote ωε = x ∈ Ω : δ(x) > ε. By the maximumprinciple applied to the equation ∆v + b(x) · ∇v = 0 where b(x) = |∇v|p−2∇v, wehave maxωε

|v| = M(ε) := max∂ωε |v|. But v ∈ C0(Ω) implies M(ε)→ 0 as ε→ 0.Consequently v ≡ 0.

Proof of Theorem 40.7. (i) Let u0 satisfy (40.8). We first claim that

limt→∞ ‖u(t)‖X = 0. (40.10)

To see this, let us first observe that φ : X → [0,∞), defined by φ(v) = ‖v‖∞ is aLyapunov functional for problem (34.5) (cf. Appendix G). Indeed, as a consequenceof (40.1), the function h(t) = ‖u(t)‖∞ is nonincreasing for t ≥ 0. Moreover, if his constant, then for each t > 0, u(·, t) achieves the value ‖u0‖∞ at some interior

point. Applying the strong maximum principle (Proposition 52.7), we infer that u isconstant, hence 0, in Ω×(0,∞). Therefore, φ is in fact a strict Lyapunov functional.Moreover, by (40.8) and parabolic estimates, u(t) is bounded in W 2,q(Ω) for eachfinite q, so that u(t) : t ≥ 1 is precompact in X . By Propositions 53.3 and 53.5,it follows that ω(u0) (in the X topology) is nonempty and consists of equilibria.Property (40.10) thus follows from Proposition 40.8.

Now the exponential decay in (40.9) follows from Remark 51.20(ii).(ii) This follows from Theorem 40.1 and assertion (i).(iii) Let Θ be the classical solution of (19.27). By Hopf’s lemma (cf. Proposi-

tion 52.1), we have Θ(x) ≥ c1δ(x) in Ω. Letting

M = ‖∇Θ‖∞ and φ = M−p/(p−1)Θ,

we find that−∆φ ≥ |∇φ|p in Ω. (40.11)

Now, assume that ‖u0‖X < c1M−p/(p−1). Consequently, u0(x) ≤ ‖u0‖Xδ(x) ≤

φ(x) in Ω. It then follows from (40.11) and the comparison principle (see Proposi-tion 52.16) that u ≤ φ in Ω× [0, T ), where T = Tmax(u0). Since u = φ = 0 on ∂Ω,we deduce that

|∇u| = −∂u

∂ν≤ −∂φ

∂ν≤ C on ST .

The result then follows from Proposition 40.3 and assertion (i).

Remarks 40.9. (a) Boundedness of global solutions. In the case p > 2 andΩ bounded, the question of boundedness of global solutions has been studied,too. For n = 1, it was shown in [38] that all global solutions are bounded in X .Consequently, they satisfy (40.9). For n ≥ 2, this is an open problem.

On the other hand, if one looks at weaker norms, an L∞ a priori estimate ofglobal solutions is provided by (40.1), and it was shown in [500] that all globalsolutions actually decay in L∞. Moreover, Theorem 40.2 implies the universalLq-bound supt≥0 ‖u(t)‖q ≤ C(Ω, p, q) for all finite q. Furthermore, under thestronger condition p > max(2, n), one actually has a universal L∞-bound of theform ‖u(t)‖∞ ≤ C(Ω, p, q)(1 + t−α) for all t > 0 and some α = α(n, p) > 0 [500].

(b) A priori estimates. Similarly to the model problem (15.1) (cf. Section 22)one can consider the question, not only of boundedness but of a priori estimates ofglobal solutions in X norm, and the related problem of continuity of the existencetime. Consider problem (34.5) with p > 2 and Ω = (0, 1), fix a nontrivial φ ∈ X+

and define

E :=λ > 0 : Tmax(λφ) <∞ and ‖u(t)‖X → 0, as t→∞

.

By Theorems 40.7(iii) and 40.2, we have λ ∈ E for λ > 0 small and λ ∈ E for λ > 0large. Therefore λ∗ := sup E ∈ (0,∞) and, due to Theorem 40.7(iii) and continuous

dependence, λ∗ ∈ E. Since all global solutions decay in X (cf. Remark 40.9(a)), itfollows that Tmax(λ∗φ) <∞. Consequently Tmax is discontinuous. Moreover globalsolutions fail to satisfy an a priori estimate of the form supt≥0 ‖u(t)‖X ≤ C(‖u0‖X)(since, by continuous dependence, this would imply a bound for ‖u(·; λ∗φ)‖X ,hence Tmax(λ∗φ) = ∞). This exhibits a similar phenomenon as for the modelproblem in dimensions n ≥ 3, which does not occur in dimensions n = 1 or 2(cf. Theorem 22.1, Theorem 28.7 for radial solutions in a ball and p > pS , and seeafter Theorem 22.13).

(c) Unbounded global solutions. If one considers the modification of problem(34.5) where a (smooth) inhomogeneous term h(x) ≥ 0 is added on RHS, thenboundedness of global solutions is still true in L∞-norm, but may fail in the Xnorm. Indeed, examples of global solutions with |∇u(x, t)| becoming unboundedon the boundary as t → ∞ have been constructed in [500] for all n ≥ 1. In [496],for a variant of problem (34.5) with n = 1, the grow-up rate of ux is determinedby techniques of matched asymptotics.

(d) Consider the situation of Theorem 40.7 in the limiting case p = 1. Then allsolutions are still global and decay exponentially, but the decay exponent can besmaller than λ1 (see [69]). On the other hand, decay no longer occurs in generalfor 0 < p < 1. Indeed, if Ω = (0, 1) for instance, it is easy to construct positivestationary solutions. We refer to [321] for results on the asymptotic behavior inthis case.

We turn to the Cauchy problem. Recall that now all solutions are global byProposition 40.5. The most complete results available concern the case of solutionswith finite mass: Unless otherwise specified, we shall assume in the rest of thissubsection that

u0 ∈ X+ ∩ L1(Rn), u0 ≡ 0. (40.12)

Under this assumption, the solution of (34.5) satisfies u ∈ C([0,∞), L1(Rn)) (thiscan be shown by arguments similar to those in the proof of (51.42) in Proposi-tion 51.16). Moreover, ‖u(t)‖1 is nondecreasing in time. This follows from∫

Rn

u(t) dx =∫

Rn

u0 dx +∫ t

0

∫Rn

|∇u(y, s)|p dy ds, (40.13)

due to Proposition 48.4(b) and the variation-of-constants formula. We may thusdefine

I∞ = limt→∞ ‖u(t)‖1 ∈ (0,∞],

and a natural question is then to determine whether the growth of mass is limitedor not, i.e., I∞ < ∞ or I∞ = ∞. It turns out that the problem involves two criticalexponents

p = 2 and p = pc := (n + 2)/(n + 1).

Recall that Gt denotes the Gaussian heat kernel, defined in (48.5).

Theorem 40.10. Consider problem (34.5) with Ω = Rn and u0 satisfying (40.12).

(i) Assume p ≥ 2. Then, for all u0, there holds I∞ <∞. Moreover,

‖u(t)− I∞Gt‖1 → 0, t→∞. (40.14)

(ii) Assume 1 < p ≤ pc. Then, for all u0, there holds I∞ = ∞.(iii) Assume pc < p < 2. Then we have I∞ < ∞ for small data (in a suitablesense), and there also exist u0 such that I∞ = ∞. Furthermore, (40.14) is satisfiedwhenever I∞ <∞.

Assertions (i) and (ii) are due to [322]. As for assertion (iii), the fact thatI∞ < ∞ under suitable smallness assumptions was proved in [145], [322], and theexistence of at least one solution such that I∞ = ∞ is due to [72]. This was nextshown to occur under suitable largeness conditions on u0 in [70]. We shall prove(i) and (ii) only. The proof of (iii) is more delicate and we refer for this to theabove mentioned articles.

Proof of Theorem 40.10(i). First observe that in view of (40.7), u satisfies

ut −∆u ≤ a|∇u|2, x ∈ Rn, t > 0

with a = ‖∇u0‖p−2∞ > 0. We use the Hopf-Cole transformation v := eau − 1. The

function v satisfies

vt −∆v = a(ut −∆u− a|∇u|2)eau ≤ 0, x ∈ Rn, t > 0.

Therefore, v(t) ≤ e−tAv0 by the maximum principle, where e−tA denotes the heatsemigroup in R

n. Using the inequalities x ≤ ex− 1 ≤ xex for x ≥ 0, it follows that

a‖u(t)‖1 ≤ ‖v(t)‖1 ≤ ‖v0‖1 ≤ ‖au0eau0‖1 ≤ aea‖u0‖∞‖u0‖1, t ≥ 0

hence I∞ < ∞. Property (40.14) is then a consequence of Lemma 20.16 (and so isthe last statement of assertion (iii)).

Proof of Theorem 40.10(ii).Case 1: n ≥ 2. Since p < n, by the Sobolev inequality and (40.13), there holds

‖u(t)‖1 ≥ C

∫ t

0

‖u(s)‖pp∗ ds, t ≥ 0, where p∗ = np/(n− p). (40.15)

Also, for |x| ≤√

t and t ≥ t0(u0) large enough, we have

u(x, t) ≥ (4πt)−n/2

∫Rn

e−|x−y|2/4tu0(y) dy

≥ (4πt)−n/2e−1

∫|y|<√

t

u0(y) dy ≥ Ct−n/2‖u0‖1.(40.16)

It follows that for all s ≥ t0(u0),

‖u(s)‖p∗p∗ ≥

∫|x|<√

s

up∗(x, s) dx ≥ C‖u0‖p∗

1 s−n2 (p∗−1),

which combined with (40.15) yields

‖u(t)‖1 ≥ C‖u0‖p1

∫ t

t0

s−np2 (1− 1

p∗ ) ds.

Since p ≤ pc, we have np2 (1− 1

p∗ ) = (n+1)p−n2 ≤ 1, hence I∞ = ∞.

Case 2: n = 1. We use the interpolation inequality

‖v‖2p−1∞ ≤ C‖v‖p−1

1 ‖vx‖pp, for all p ≥ 1 and v ∈ L1 such that vx ∈ Lp,

(40.17)which is a consequence of

|v(x)|(p−1)/pv(x) =2p− 1

p

∫ x

−∞|v|(p−1)/pvx dy

and of Holder’s inequality. Since ‖u(t)‖1 is nondecreasing, it follows from (40.13)and (40.17) that

‖u(t)‖p1 ≥ ‖u(t)‖p−1

1

∫ t

0

‖ux(s)‖pp ds

≥∫ t

0

‖u(s)‖p−11 ‖ux(s)‖p

p ds ≥ C

∫ t

0

‖u(s)‖2p−1∞ ds.

But (40.16) implies ‖u(t)‖∞ ≥ C‖u0‖1t−1/2 for t ≥ t0(u0) large enough, hence

‖u(t)‖p1 ≥ C‖u0‖2p−1

1

∫ t

t0

s−p+ 12 ds.

Since p ≤ pc = 3/2, we conclude that I∞ = ∞.

Remarks 40.11. (a) Nonlinear asymptotic behaviors. Theorem 40.10 showsthat when p ≥ 2, or pc < p < 2 and u0 is small, then I∞ <∞ and the asymptoticbehavior is dominated by the diffusion. When I∞ = ∞, other behaviors are known.To describe this briefly, first observe that, since ‖u(t)‖∞ is nonincreasing in timedue to (40.1), we may set

N∞ := limt→∞ ‖u(t)‖∞ ∈ [0,∞).

It was proved in [70] that if (and only if)

N∞ > 0, (40.18)

then I∞ = ∞ and u behaves like the viscosity solution z of the pure Hamilton-Jacobi equation zt = |∇z|p, with initial data N∞χ0. More precisely,

limt→∞ ‖u(t)− z(t)‖∞ = 0, where z(x, t) =

(N∞ − c(p)

( |x|t1/p

)p/(p−1))+.

In the range 1 < p ≤ pc, property (40.18) is true for all nontrivial (not necessarilyintegrable) u0 ∈ X+, see [251]. The situation is different in the range pc < p < 2:property (40.18) holds under a suitable largeness condition on u0 [70], but anexample of a solution such that

I∞ =∞ and N∞ = 0 (40.19)

has been constructed in [72]. This solution is self-similar, of the form

u(x, t) = (t + 1)−kV( x√

t + 1

), k =

2− p

2(p− 1),

where the profile V ∈ L1(Rn) decays exponentially at infinity (cf. Remark 15.4(ii)for an analogue in the model problem (15.1)). This corresponds to an intermediatebehavior involving a balance between the diffusion and the nonlinear term. It isunknown whether this self-similar solution is unique, nor if there exist solutionssatisfying (40.19) other than self-similar.

(b) For estimates on I∞ (if finite) or on the growth rate of ‖u(t)‖1 otherwise,see [322], [70], [251]. An alternative proof of Theorem 40.10(i) based on multiplierarguments (instead of the Hopf-Cole transformation) is also given in [322].

(c) Estimates similar to (40.14) are also true for other Lq-norms [70]. Namely,for every q ∈ [1,∞], there holds

t(n/2)(1−(1/q))‖u(t)− I∞Gt‖q → 0, t→∞.

(d) For general solutions of (34.5) (assuming only u0 ∈ X+ but not u0 ∈L1(Rn)), some results on the asymptotic behavior can be found in [71], [252],[251], [491]. In particular it was shown in [71], [252] by Bernstein-type techniques,that any solution satisfies the global gradient estimate

|∇u(x, t)| ≤ C(p)‖u0‖1/p∞ t−1/p, x ∈ R

n, t > 0

(hence ‖∇u(t)‖∞ → 0 as t→∞).(e) The exponents p = 2 and p = pc are also critical in the local existence-

uniqueness theory of problem (34.5) with irregular initial data u0; see [27], [71],[72].

40.3. Space profile of gradient blow-up

In this subsection we study the space profile of GBU of solutions to (34.5) forp > 2. We shall restrict ourselves to one space dimension.

Definition 40.12. Let Ω ⊂ Rn and consider problem (34.5). We say that x0 ∈ Ω

is a GBU point (in finite or infinite time) if there exist sequences tj → Tmax(u0)and xj → x0 such that |ux(xj , tj)| → ∞.

In order to formulate our results, it is convenient to introduce the steady statesof (34.5) for n = 1. They will be useful again in the study of the time rate of GBU;see the proof of Theorem 40.19 in the next subsection. To describe these steadystates, let us denote

U(x) := dp x(p−2)/(p−1), U ′(x) = d′p x−1/(p−1), x > 0, (40.20)

where dp = (p − 2)−1(p − 1)(p−2)/(p−1) and d′p = (p − 1)−1/(p−1). The functionU ∈ C([0,∞))∩C1((0,∞)) is a “singular” steady state. Namely, it is a solution of

V ′′ + V ′p = 0, x > 0, V (0) = 0, (40.21)

which satisfies Ux(0) = ∞. Next, for each λ > 0, we put

Uλ(x) := U(x + λ)− U(λ). (40.22)

Each Uλ ∈ C1([0,∞)) also solves (40.21). Moreover we have U ′λ(x) → ∞, as

x → 0+ and λ→ 0+, and Uλ(x) → U(x), uniformly for x ∈ [0, 1], as λ→ 0+.

Our first result gives bounds on ux away from x = 0 and 1. This shows inparticular that GBU may occur only on the boundary.

Proposition 40.13. Consider problem (34.5) with p > 2 and Ω = (0, 1). Letu0 ∈ X+ and 0 < t0 < T := Tmax(u0). There exists C1 > 0 such that, for allt0 ≤ t < T ,

ux(x, t) ≤ U ′(x) + C1x, 0 < x ≤ 1 (40.23)

andux(x, t) ≥ −U ′(1 − x)− C1(1− x), 0 ≤ x < 1, (40.24)

where U is defined by (40.20). In particular x = 0 and x = 1 are the only possibleGBU points.

The next result gives a precise description of the spatial profile around a GBUpoint. It is essentially due to [138] (where a slightly different problem, with non-homogeneous boundary value at x = 1, was actually studied).

Theorem 40.14. Consider problem (34.5) with p > 2 and Ω = (0, 1). Let u0 ∈X+ and assume that T := Tmax(u0) <∞.(i) For each x ∈ (0, 1), the limits

u(x, T ) := limt→T

u(x, t) and ux(x, T ) := limt→T

ux(x, t) exist and are finite.

Moreover, the first (resp., second) limit is uniform (resp., locally uniform) forx ∈ (0, 1).(ii) If 0 is a GBU point, then

limt→T

ux(0, t) =∞ (40.25)

and there exist C1, C2 > 0 such that

U(x) − C1x ≤ u(x, T ) ≤ U(x) + C2x2, 0 < x ≤ 1/2 (40.26)

andU ′(x) − C1 ≤ ux(x, T ) ≤ U ′(x) + C2x, 0 < x ≤ 1/2, (40.27)

where U is defined by (40.20). Similar estimates hold if 1 is a GBU point.

As a preliminary to the proofs, we need the following simple properties of thetime-derivative ut. (They are valid without restriction on n and will be used alsoin the next subsection.) We first note that ut ∈ C2,1(QT ) by parabolic regularityresults, and that ut ∈ BC(Ω×[t0, t1]), 0 < t0 < t1 < T , due to (35.3). The functionw := ut satisfies

wt −∆w = a(x, t) · ∇w, x ∈ Ω, 0 < t < T,

w = 0, x ∈ ∂Ω, 0 < t < T,

(40.28)

wherea(x, t) = p|∇u|p−2∇u. (40.29)

As an immediate consequence of (40.28) and of the maximum principle, we have:

Lemma 40.15. Consider problem (34.5) with p > 1 and u0 ∈ X+, and let 0 <t0 < T := Tmax(u0). There exists C1 > 0 such that

|ut| ≤ C1, x ∈ Ω, t0 ≤ t < T. (40.30)

Proof of Proposition 40.13. Fix 0 < t0 < t < T and let y(x) = (ux(x, t) −C1x)+, where C1 is given by Lemma 40.15. The function y satisfies

y′ + yp = (uxx − C1)χux>C1x + (ux − C1x)p+, for a.e. x ∈ (0, 1).

For each x such that ux(x, t) > C1x, we have (y′+yp)(x) ≤ (uxx−C1+|ux|p)(x) ≤ 0by (40.30). Therefore, we have y′ + yp ≤ 0 a.e. on (0, 1). By integration, it followsthat y(x) ≤ ((p− 1)x)−

1p−1 , hence (40.23).

As for (40.24), it follows by applying (40.23) to the function u(1− x, t), whichsatisfies the same equation.

For the proof of Theorem 40.14 we need the following lemma, which will providea lower bound on the blow-up profile of ux at x = 0.

Lemma 40.16. Consider problem (34.5) with p > 2 and Ω = (0, 1). Let u0 ∈ X+

and 0 < t0 < T := Tmax(u0). There exists C3 > 0 such that, for all t0 ≤ t < T ,[(ux(x, t))+ + C3

]1−p ≤[ux(0, t) + C3

]1−p + (p− 1)x, 0 ≤ x ≤ 1. (40.31)

Proof. Fix t ∈ [t0, T ) and let

z(x) = (ux(x, t))+ + C1/p1 ,

where C1 is given by Lemma 40.15. The function z satisfies

z′ + zp = uxxχux>0 +[(ux(x, t))+ + C

1/p1

]p ≥ (uxx + |ux|p)χux>0 + C1 ≥ 0

a.e. on (0, 1) by (40.30). By integration, it follows that z1−p(x) ≤ z1−p(0)+(p−1)x.Using ux(0, t) ≥ 0, we obtain (40.31) with C3 = C

1/p1 .

Proof of Theorem 40.14. (i) It follows from Proposition 40.13 that ux(·, t)is bounded in L1(0, 1). This along with (40.1) implies u(·, t) : t ∈ (0, T ) isprecompact in C([0, 1]). Using Lemma 40.15, we deduce that limt→T u(x, t) exists,uniformly for x ∈ [0, 1].

Now fix t0 ∈ (0, T ). By Proposition 40.13 and Lemma 40.15, we deduce thatux, uxx ∈ L∞((ε, 1−ε)×(t0, T )) for each ε ∈ (0, 1). Therefore, ux(·, t) : t ∈ (t0, T )is precompact in C((0, 1)). Since uxt− uxxx = p|ux|p−2uxuxx, parabolic estimatesimply uxt ∈ Lq((ε, 1−ε)×(t0, T )) for each ε ∈ (0, 1) and each finite q. Consequently,ux ∈ BUCα([ε, 1− ε]× [t0, T )) for each ε ∈ (0, 1) and some α ∈ (0, 1). We deducethat limt→T ux(x, t) exists, locally uniformly for x ∈ (0, 1).

(ii) The upper estimates in (40.26), (40.27) follow from Proposition 40.13. Toshow the lower estimates, let us first note that, by assumption, there exist se-quences tj → T and xj → 0 such that |ux(xj , tj)| → ∞, hence ux(xj , tj)→∞ dueto (40.24). Moreover, by Lemma 40.15 and (34.5), we have uxx ≤ C1, hence

ux(0, t) ≥ ux(x, t) − C1x, (x, t) ∈ [0, 1]× [t0, T ). (40.32)

It follows that ux(0, tj) →∞. Put εj := (ux(0, tj)+C3)1−p → 0, where C3 is fromLemma 40.16. By that lemma, there exists η ∈ (0, 1) such that, for j large,

ux(x, tj) ≥ (εj + (p− 1)x)−1/(p−1) − C3, 0 < x < η,

hence

u(x, tj) ≥1

p− 2[(εj + (p− 1)x)(p−2)/(p−1) − ε

(p−2)/(p−1)j

]− C3x, 0 < x < η.

Letting j → ∞, and since we already know that the limits in assertion (i) exist,we obtain the lower estimates in (40.26), (40.27) (η can be replaced by 1/2 byenlarging the constant C1).

Finally, using (40.32), we may write, for each x ∈ (0, 1),

lim inft→T

ux(0, t) ≥ lim inft→T

ux(x, t)− C1x ≥ U ′(x)− C1(x + 1),

and (40.25) follows by letting x→ 0.

Remark 40.17. The analogue of the upper estimate (40.23) is still true in higherdimensions. Namely, by using Bernstein-type techniques, it is shown in [500] thatany solution of (34.5) with p > 2 satisfies

|∇u(x, t)| ≤ C1δ−1/(p−1)(x) + C2 in Ω× [0, Tmax(u0)),

with C1 = C1(p, n) > 0 and C2 = C2(u) > 0.

40.4. Time rate of gradient blow-up

We now study the time rate of GBU of solutions to (34.5) for p > 2, i.e.: the speedof divergence of ‖∇u(t)‖∞. We begin with lower estimates. The following theoremis due to [263].

Theorem 40.18. Consider problem (34.5) with p > 2 and Ω = Rn. Let u0 ∈ X+

and assume that T := Tmax(u0) < ∞. Then there exists C > 0 such that

sups∈[0,t]

‖∇u(s)‖∞ ≥ C(T − t)−1/(p−2), t→ T. (40.33)

In one space dimension, we have the following more precise result from [138](see also Remark 40.23 below).

Theorem 40.19. Consider problem (34.5) with p > 2, n = 1 and Ω = (0, 1). Letu0 ∈ X+ and assume that T := Tmax(u0) < ∞. Then there exists C > 0 such that

‖ux(t)‖∞ ≥ C(T − t)−1/(p−2), t→ T. (40.34)

Remarks 40.20. (i) Non self-similar GBU rate. The lower estimate (40.33)implies in particular that the GBU rate does not correspond to the one suggestedby the self-similar invariance of the problem. Indeed, letting k = (p−2)/(2(p−1)),the scaling transformations

Sλ : u → uλ(x, t) := λ2ku(λ−1x, λ−2t), λ > 0,

leave invariant the equation in (34.5). This might allow for the existence of back-ward self-similar (classical) solutions of the form

w(x, t) = (T − t)kV( x√

T − t

)(40.35)

(note that forward self-similar solutions in Rn exist for some p < 2, cf. Re-

mark 40.11(a)). Now if there exists a nontrivial solution w of the form (40.35),say, on Ω = (0,∞) with zero boundary condition at x = 0, and if ∇V ∈ L∞, then

w will exhibit the GBU self-similar rate ‖∇w(t)‖∞ = C(T − t)−1/(2(p−1)). How-ever, since 1/(p− 2) > 1/(2(p− 1)), Theorem 40.18 shows that no such solutionsw exist and that the exponent of the GBU rate is always greater than that ofthe self-similar rate. A similar situation has been encountered for the supercriticalmodel problem (cf. Section 23) and also for problem (38.3).

(ii) A rough argument involving the variation-of-constants formula would alsogive a lower estimate (T − t)−1/(2(p−1)).

The upper blow-up rate estimate for problem (34.5) is still an open question.However, some results are known for the closely related one-dimensional problem:

ut − uxx = |ux|p + λ, 0 < x < 1, t > 0,

u = 0, x ∈ 0, 1, t > 0,

u(x, 0) = u0(x), 0 < x < 1,

⎫⎪⎬⎪⎭ (40.36)

with p > 2, λ > 0 and u0 ∈ X+. Note that the local solution of (40.36) is nonneg-ative and uniformly bounded on finite time intervals (since u(x, t) := ‖u0‖∞ + λtis a supersolution). Moreover, as a consequence of the proof of Theorem 40.2, gra-dient blow-up occurs whenever λ > λ0(p) or ‖u0‖q ≥ C(p, q) for some q ∈ [1,∞),where λ0(p) and C(p, q) are suitable positive constants.

Theorem 40.21. Consider problem (40.36) with p > 2 and λ > 0. Let u0 ∈X+ ∩C2([0, 1]) be symmetric with respect to x = 1/2 and satisfy

u0,xx + |u0,x|p + λ ≥ 0 in [0, 1]. (40.37)

If T := Tmax(u0) <∞, then there exists C > 0 such that

‖ux(t)‖∞ ≤ C(T − t)−1/(p−2), t→ T. (40.38)

Theorem 40.21 is a variant of a result of [263], where the authors consideredthe equation in (34.5) under inhomogeneous boundary conditions for n = 1. Forthat problem the upper GBU rate estimate was first conjectured in [138] on thebasis of numerical simulations.

Remarks 40.22. (i) Assumption (40.37) guarantees that the solution is nonde-creasing in time. However, analogous assumption cannot be satisfied for problem(34.5). Indeed if u0 ∈ X+ ∩ C2([0, 1]) verifies (40.37) with λ = 0, then u0 ≡ 0(this follows for instance from the maximum principle). On the other hand, it isunknown if Theorem 40.21 (and the corresponding result in [263]) remains truewithout assumption (40.37) or in higher space dimensions.

(ii) Estimate (40.38) is sharp. In fact, under the assumptions of Theorem 40.21,the lower estimate (40.34) follows from simple modifications of the proof of The-orem 40.18, along with (40.51) below.

We first give (a variant of) the proof of Theorem 40.18 from [263]. It relies onlinear regularity estimates applied to the equation for ut.

Proof of Theorem 40.18. Denote

m(t) := supΩ×[T/2,t]

|∇u| = maxs∈[T/2,t]

‖∇u(s)‖∞, T/2 ≤ t < T (40.39)

(note that m(t) and ‖∇u(t)‖∞ are continuous due to (51.27)). In this proof, C willdenote positive constants, independent of t ∈ (T/2, T ), which may change fromline to line and may depend on u.

Step 1. We claim that w := ut satisfies

‖∇w(t)‖∞ ≤ Cmp−1(t), T/2 < t < T. (40.40)

Let t ∈ (T/2, T ), s ∈ (T/4, t), and put K = supσ∈[0,t−s] σ1/2‖∇w(s+σ)‖∞. For

τ ∈ (0, t − s), in view of (40.28), (40.29) and the variation-of-constants formula,we have

w(s + τ) = e−τAw(s) +∫ τ

0

e−(τ−σ)A(a · ∇w)(s + σ) dσ.

Using Proposition 48.7, Lemma 40.15, and the fact that∫ τ

0 (τ − σ)−1/2σ−1/2 dσ =∫ 1

0(1− z)−1/2z−1/2 dz, it follows that

‖∇w(s + τ)‖∞ ≤ Cτ−1/2‖w(s)‖∞ + C

∫ τ

0

(τ − σ)−1/2‖a · ∇w(s + σ)‖∞ dσ

≤ Cτ−1/2 + Cmp−1(t)K.

Multiplying by τ1/2 and taking the supremum for τ ∈ [0, t− s], we obtain

K ≤ C + C(t− s)1/2mp−1(t)K.

Now choosing s = t − (1/4)min(T, (Cmp−1(t))−2

)∈ (T/4, t), we obtain K ≤ 2C,

hence‖∇w(t)‖∞ ≤ 2C(t− s)−1/2 ≤ 4C max(T−1/2, Cmp−1(t)).

Since m is positive nondecreasing, this implies Claim (40.40).Step 2. We next claim that m is locally Lipschitz on (T/2, T ) and that

m′(t) ≤ Cmp−1(t), for a.e. t ∈ (T/2, T ). (40.41)

Let T/2 < t < s < T . For any τ ∈ [t, s] and x ∈ Ω, it follows from themean-value inequality and (40.40) that

|∇u(x, τ)−∇u(x, t)| ≤ (τ − t) supΩ×[t,τ ]

|∂t∇u| ≤ C(τ − t)mp−1(τ)

hence

|∇u(x, τ)| ≤ |∇u(x, t)|+ C(τ − t)mp−1(τ) ≤ m(t) + C(s− t)mp−1(s).

Taking supremum for (x, τ) over Ω× [t, s], we get

0 ≤ m(s)−m(t) ≤ C(s− t)mp−1(s).

Since m is continuous, the claim follows.Finally, integrating (40.41) over (t, s) with T/2 < t < s < T , and using m(s)→

∞ as s → T , we infer that

m(t) ≥ C(T − t)−1/(p−2), (40.42)

which implies estimate (40.33).

We next give the proof of Theorem 40.19, based on (a simplification of) the ideafrom [138]. It relies on a completely different argument, involving the intersectionsof the solution with the singular steady state.

Proof of Theorem 40.19. Recall that the steady states U and Uλ are definedin (40.20) and (40.22). Due to Proposition 40.3, we may assume, without loss ofgenerality, that

lim supt→T

ux(0, t) = ∞. (40.43)

Fix t0 ∈ [T/2, T ) and let

x0 := supx ∈ (0, 1] : ux(·, t0) 0 small, x0 is well defined and x0 > 0.On the other hand, by definition, we have ux(x, t0) 0 small. (40.44)

We claim that x0 ∈ (0, 1), hence

ux(x0, t0) = U ′(x0). (40.45)

Indeed, otherwise x0 = 1, so that (40.44) implies u ≤ Uλ in [0, 1] × [t0, T ) forλ > 0 small, due to the comparison principle. Therefore ux(0, t) ≤ U ′

λ(0) in [t0, T ),contradicting (40.43).

We next claim thatsup

t∈[t0,T )

u(x0, t) ≥ U(x0). (40.46)


Suppose the contrary. Then, for all λ > 0 small, we have u(x0, t) ≤ Uλ(x0)in [t0, T ). By (40.44) and the comparison principle, we deduce that u ≤ Uλ in[0, x0]× [t0, T ), leading again to a contradiction.

Now, using Lemma 40.15 and (40.46), we get

C1(T − t0) ≥∫ T

t0

ut(x0, t) ≥ U(x0)− u(x0, t0) =∫ x0

0

(U ′(x) − ux(x, t0)) dx.

On the other hand, by (40.45), there clearly exists x1 ∈ (0, x0] such that U ′(x1) =max[0,1] ux(·, t0). Since U ′(x)− ux(x, t0) > 0 on (0, x0) by the definition of x0, weobtain

C1(T − t0) ≥∫ x1

0

(U ′(x)− ux(x, t0)) dx

≥ U(x1)− x1U′(x1) =

U ′(x1)2−p

(p− 1)(p− 2)≥ ‖ux(t0)‖2−p

∞(p− 1)(p− 2)

,

and the conclusion follows.

Remark 40.23. More precise lower estimate. In addition to the hypothesesof Theorem 40.19, assume that u0 ∈ C2([0, 1]) and denote

C1 = max[0,1]

(u0,xx + |u0,x|p).

Then the proof of Theorem 40.19 provides the value C =((p−1)(p−2)C1

)−1/(p−2)

for the constant in (40.34).

Finally, we give the proof of Theorem 40.21, based on the ideas in [263], whichrelies on the application of the maximum principle to a suitable auxiliary function.Note that this function (cf. w below) is quite different from the function J usedin the proof of Theorem 24.1.

Proof of Theorem 40.21. We consider the parabolic operator

Lφ := φt − φxx − p|ux|p−2uxφx.

For σ ∈ (0, 1) and t0 ∈ (0, T ) to be chosen later, we introduce the auxiliary function

w(x, t) :=(1 +

1mσ(t)

)(1− ux

m(t)

), x ∈ [0, 1], t ∈ [t0, T ),

wherem(t) := max

(x,t)∈[0,1]×[t0,t]|ux(x, t)| → ∞, as t→ T. (40.47)

Step 1. We shall show that for suitable t0 ∈ (0, T ) and C > 0, there holds

w + u ≤ Cut in (0, 1)× (t0, T ). (40.48)

We may assume m(t) ≥ 1 without loss of generality. Also, by the proof ofTheorem 40.18, m is locally Lipschitz on (T/2, T ) and (40.41) is satisfied. A directcomputation shows that

Lw = − σm′

mσ+1

(1− ux

m

)+(1 +

1mσ

)uxm′

m2. (40.49)

Since m′ ≥ 0 a.e., we have, in case |ux(x, t)| < σσ+2m1−σ(t),

Lw =m′

mσ+1

(−σ + (σ + 1)

ux

m+

ux

m1−σ

)≤ m′

mσ+1

(−σ + (σ + 2)

|ux|m1−σ

)≤ 0

(a.e. in t). On the other hand, if |ux(x, t)| ≥ σσ+2m1−σ(t), then by (40.49) and

(40.41), we have

Lw ≤(1 +

1mσ

)uxm′

m2≤ C|ux|

mp−1

m2≤ C

m

(σ + 2σ

)(p−2)/(1−σ)

|ux|(p−1−σ)/(1−σ).

We now choose σ = 1/(p − 1), so that (p − 1 − σ)/(1 − σ) = p. Thus, takingC := C(σ+2

σ )(p−2)/(1−σ) and using (40.47), we obtain, for t0 close to T ,

L(w +u) ≤ C

m|ux|p− (p−1)|ux|p +λ ≤ −p− 1

2|ux|p +λ, a.e. in (0, 1)× [t0, T ).

If |ux(x, t)|p ≥ 2λ/(p − 1), then L(w + u) ≤ 0, whereas w(x, t) ≥ 1/2 otherwise(for t0 close to T ). In all cases we thus have L(w + u) ≤ 2λ(w + u), hence

L(e−2λt(w + u)

)≤ 0 = Lut, a.e. in (0, 1)× [t0, T ). (40.50)

Due to our assumptions on u0, u is symmetric with respect to x = 1/2, andwe have ut ≥ 0 (and ut ≡ 0) in (0, 1)× (0, T ) by Proposition 52.19. In particulart → ux(0, t) is nonnegative and nondecreasing, and it follows from the proof ofProposition 40.3 that

m(t) = ‖ux(t)‖∞ = ux(0, t), t0 ≤ t < T, (40.51)

by taking t0 closer to T if necessary. Consequently,

[w + u](x, t) = 0 = ut(x, t), x ∈ 0, 1, t0 ≤ t < T. (40.52)

On the other hand, the strong and Hopf maximum principles guarantee that

ut(x, t0) > 0, 0 < x < 1, utx(0, t0) > 0, utx(1, t0) < 0.

In particular there exists C > 0 such that[e−2λt0(w + u)− Cut

](x, t0) ≤ 0, 0 < x < 1. (40.53)

Using (40.50), (40.52), (40.53), and the maximum principle (under the form ofProposition 52.8), we deduce e−2λt(w + u) ≤ Cut in (0, 1)× [t0, T ), hence (40.48).

Step 2. As a consequence of (40.48) and (40.51), we have

Cuxt(0, t) = limx→0+

Cut(x, t)x

≥ limx→0+

[w + u](x, t)x

≥ limx→0+

w(x, t)x

= wx(0, t) = −(1 +

1mσ(t)

)uxx(0, t)m(t)

≥ |ux(0, t)|pm(t)

= up−1x (0, t).

By integration, we obtain

ux(0, t) ≤ C(T − t)−1/(p−2), t→ T,

which proves the result.

Remarks 40.24. (a) Boundary layer. Under the assumptions of Theorem 40.21,for any K > 0, there exists c = c(K) > 0 such that

ux(x, t) ≥ c(T − t)−1/(p−2) for 0 < x < K(T − t)(p−1)/(p−2)

and t close to T (cf. [263]). This boundary layer estimate follows from the proofof Lemma 40.16, estimate (40.34) (cf. Remark 40.22(ii)) and (40.51).

(b) Nonsymmetric initial data. In Theorem 40.21, the symmetry assumptionon u0 can be removed. To show this, assuming that x = 0 (resp. x = 1) is aGBU point, one has to replace the interval [0, 1] by [0, 1/2] (resp. [1/2, 1]) in theproof of Theorem 40.21 (and in particular in the definition (40.47) of m(t)). Onethen uses the fact that ux is bounded away from the boundary (cf. the proof ofTheorem 40.14) and that estimate (40.41) remains true (this follows from simplemodifications of the proof of Theorem 40.18, using wφ instead of w in Step 1,where φ = φ(x) is a cut-off function equal to 1 on [0, 1/2] and to 0 near x = 1).

On the other hand, by similar arguments, one can show that (under the hypothe-ses of Theorem 40.21 without the symmetry assumption) there holds |ux(x, t)| ≥C(T − t)−1/(p−2) as t→ T , for each GBU point x ∈ 0, 1.

(c) Continuation after GBU. For some results on continuation after GBU,see [193], [57], [201]. These references contain examples where all solutions canbe continued in some sense after GBU (and not only threshold solutions, like inL∞-blow-up — cf. Proposition 27.7 and Remark 27.8(c)).

41. An example of interior gradient blow-up

In the previous section we studied the phenomenon of gradient blow-up on theboundary. The aim of this section is to provide a simple example of a differentbehavior, namely: interior gradient blow-up.

Consider the following problem:

ut − uxx = |u|m−1u|ux|p, − 1 < x < 1, t > 0,

u(±1, t) = A±, t > 0,

u(x, 0) = u0(x), − 1 < x < 1,

⎫⎪⎬⎪⎭ (41.1)

where p > 2, m ≥ 1, A− < 0 < A+ and

u0 ∈ C1([−1, 1]), with u0(−1) = A− ≤ u0 ≤ A+ = u0(1) in [−1, 1]. (41.2)

Unlike in problem (34.5), the nonlinearity here changes sign, and this is the key fea-ture that will allow for interior GBU rather than boundary GBU (see Remark 41.4below).

The examples in Remark 51.11 guarantee that (41.1) is locally well-posed (ob-serve for instance that (41.1) can be converted to a problem with homogeneousboundary conditions by the change of unknown v(x, t) = u(x, t)−φ(x), where φ isan affine function such that φ(±1) = A±). By the maximum principle and (41.2),we immediately obtain

A− ≤ u(x, t) ≤ A+, −1 ≤ x ≤ 1, 0 ≤ t < Tmax(u0). (41.3)

Therefore Tmax(u0) < ∞ guarantees that GBU occurs (i.e. (40.2)).

Theorem 41.1. Consider problem (41.1) with p > 2, m ≥ 1. There exists L =L(m, p) > 0 such that, if max(A+, |A−|) > L, then Tmax(u0) < ∞ for any u0

satisfying (41.2).

Theorem 41.1 is (a variant of) a result from [31]. The original proof was basedon the construction of appropriate traveling wave sub- and supersolutions. We herepresent a simpler proof based on a multiplier argument similar to that in the proofof Theorem 40.2.

Proof. Let k = (p + 2m)/(p− 2). In what follows, C and C1 denote any positiveconstant depending only on p, m. For all t ∈ (0, Tmax(u0)), multiplying (41.1) by|u|k−1u and integrating by parts, we get

d

dt

∫ 1

−1

|u|k+1

k + 1dx =

[ux|u|k−1u

]1−1− k

∫ 1

−1

|ux|2|u|k−1 dx +∫ 1

−1

|ux|p |u|m+k dx.

(41.4)

41. An example of interior gradient blow-up 375

Next, by Holder’s inequality, we have∫ 1

−1

|ux|p |u|m+k dx = C

∫ 1

−1

∣∣(|u|(m+k)/pu)x

∣∣p ≥ C∣∣∣∫ 1

−1

(|u|(m+k)/pu

)x

∣∣∣p= C

(A

(p+m+k)/p+ + |A−|(p+m+k)/p

)p≥ CLp+m+k.

(41.5)

Moreover, since p(k − 1)/2 = m + k, Young’s inequality yields

k

∫ 1

−1

|ux|2|u|k−1 dx ≤ 12

∫ 1

−1

|ux|p|u|m+k dx + C. (41.6)

On the other hand, (41.3) implies

ux(±1, t) ≥ 0. (41.7)

Combining (41.4)–(41.7), and taking L = L(p, m) large enough, we obtain

d

dt

∫ 1

−1

|u|k+1 dx ≥ CLp+m+k − C1 ≥ 1.

Integrating and using (41.3), it follows that

t ≤∫ 1

−1

|u|k+1 dx ≤ 2(max(A+, |A−|)

)k+1,

hence Tmax(u0) < ∞.

Remark 41.2. It can be shown that if A+ and |A−| are small enough, then thereexist stationary, hence global solutions (see [31]). In this case the argument of theabove proof still shows that GBU occurs for suitably large initial data.

The next result asserts that, for m = 1 and a suitable class of initial data, GBUoccurs at a single interior point, namely x0 = 0. Moreover, an upper estimate isgiven for the final profile. A much more general result of interior GBU was provedin [31] (see Remark 41.4 below). However the proof therein is more delicate.

Theorem 41.3. Consider problem (41.1) with p > 2, m = 1, and A± = ±Awith A > L, where L is defined in Theorem 41.1. Let u0 ∈ C2([−1, 1]) be an oddfunction satisfying

u0,x ≥ 0, u0,xx ≤ 0 in [0, 1], u0(1) = A, u0,xx(1) + A|u0,x(1)|p = 0.

Then, there holds T := Tmax(u0) <∞,

limt→T

ux(0, t) =∞ (41.8)

and0 ≤ ux(x, t) ≤ A|x|−1, 0 < |x| < 1, 0 < t < T. (41.9)

Proof. By local uniqueness, we have u(−x, t) = −u(x, t). Let v = ux and w = uxx.By parabolic regularity results, we have v, w ∈ C2,1(QT ) ∩ C(QT ). We compute

vt − vxx = (u|v|p)x = |v|pv + pu|v|p−2vw.

Due to (41.7) and u0,x ≥ 0 in [0, 1], the maximum principle implies v ≥ 0. Differ-entiating again in x, we get

wt − wxx = (p + 1)vpw + p(uvp−1w)x = a(x, t)w + b(x, t)wx,

where a = (2p + 1)vp + p(p − 1)uvp−2w and b = puvp−1. Moreover w(1, t) =−uvp(1, t) ≤ 0 and, since u(0, t) = 0, we have w(0, t) = −uvp(0, t) = 0. We thusinfer from the maximum principle that

uxx = w ≤ 0, 0 ≤ x ≤ 1, 0 ≤ t < T.

For all 0 < x ≤ 1 and 0 ≤ t < T , it follows that ux(x, t) ≤ ux(0, t). Therefore,ux(0, t) = ‖ux(t)‖∞, hence (41.8). On the other hand, by concavity, we have

A ≥ u(x, t)− u(0, t) ≥ xux(x, t), 0 < x ≤ 1, 0 ≤ t < T,

hence (41.9).

Remark 41.4. In fact, it was proved in [31] that for any m ≥ 1 and any ini-tial data as in Theorem 41.1, interior GBU occurs, in the sense that ux remainsbounded close to the boundary. Moreover, GBU may occur only at points “whereu changes sign”; more precisely, for x0 ∈ (−1, 1), if u remains bounded away from0 in a neighborhood of x0 as t → T , then ux remains bounded near x0. The proofis based on Bernstein-type arguments.

Chapter V

Nonlocal Problems

42. Introduction

In this chapter, we study various problems with nonlocal nonlinearities. The equa-tions that we consider involve nonlocal terms taking the form of an integral inspace, or in time. These terms may also be combined with local ones, either in anadditive or in a multiplicative way.

In Sections 43–44, we consider several problems with space integrals from thepoint of view of global existence, blow-up and a priori estimates. In particular,we study in some detail the asymptotic behavior of blowing-up solutions. Thephenomenon of global blow-up appears as a typical feature of nonlocal problems.As an example of applied interest, we discuss the thermistor problem, which arisesin the modeling of Ohmic heating. Fujita-type results for problems with spaceintegrals are next described in Section 45. Finally, Section 46 is devoted to nonlocalproblems (in time) with memory terms.

Throughout this chapter we will only consider nonnegative solutions, except inSubsection 44.4. Unless otherwise specified, each of the problems below is locallywell-posed for (nonnegative) L∞ initial data, and the (nonnegative) solution enjoysthe regularity property (16.2) (see Example 51.13 and cf. Example 51.9). Also,we have the usual blow-up alternative in L∞ (cf. Proposition 16.1). As for thecomparison principle, more care is necessary when considering nonlocal problems,since it may be valid for certain problems and fail for some others. This will bemade precise whenever necessary.

43. Problems involving space integrals (I)

We consider the following problem

ut −∆u =∫

Ω

up(y, t) dy − kuq, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (43.1)

where Ω ⊂ Rn is a bounded domain, p > 1, q ≥ 1, k ≥ 0 and u0 ∈ L∞(Ω), u0 ≥ 0.

Note that problem (43.1) with k = 0 is maybe the simplest analogue of the model

378 V. Nonlocal Problems

problem (15.1) for which the nonlocal, superlinear source term is given by a spaceintegral.


When k > 0, (43.1) involves a competition between nonlocal source and localdamping terms. A basic question is to determine the conditions for global exis-tence or nonexistence of solutions. An answer is provided by the following theorem[525], which in particular shows that the value q = p represents a critical blow-up exponent. It also contains some information concerning the global asymptoticbehavior.

Theorem 43.1. Consider problem (43.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0,and 0 ≤ u0 ∈ L∞(Ω).

(i) Assume p > q or p = q and k < |Ω|. Then:(i1) there exists u0 such that Tmax(u0) < ∞;(i2) the trivial solution is locally exponentially stable, i.e.: for ‖u0‖∞ small

enough, u is global and ‖u(t)‖∞ decays exponentially to 0 as t→∞.

(ii) Assume p = q and k ≥ |Ω|. Then the trivial solution is globally exponen-tially stable, i.e.: all solutions of (43.1) are global, bounded and ‖u(t)‖∞ decaysexponentially to 0 as t→∞.

(iii) Assume p < q and k > 0. Then:(iii1) all solutions of (43.1) are global and bounded;(iii2) if k is sufficiently large, then the trivial solution is globally exponentially

stable;(iii3) if k is sufficiently small, then there exist positive stationary solutions.

Proof. (i1) We first prove the existence of blowing-up solutions in the case p = qand k < |Ω|. Fix a subdomain Ω′ ⊂⊂ Ω, such that δ := (|Ω′| − k)/2 > 0. Thereexists ψ ∈ D(Ω) such that

ψ = 1 in Ω′ and 0 ≤ ψ ≤ 1 in Ω

and we have ∫Ω

ψ dx ≥ |Ω′| = k + 2δ and ∆ψ ≥ −K

for some K > 0. Let y(t) =∫Ω u(t)ψ dx. Multiplying (43.1) by ψ, integrating by

parts and using Holder’s inequality, we obtain

y′(t) =∫

Ω

u∆ψ dx +∫

Ω

ψ dx

∫Ω

up dx− k

∫Ω

upψ dx ≥ −K

∫Ω

u dx + 2δ

∫Ω

up dx

≥ δ

∫Ω

up dx +[δ|Ω|1−p

(∫Ω

u dx)p−1

−K] ∫

Ω

u dx.

43. Problems involving space integrals (I) 379

Setting g(t) = δ|Ω|1−p(∫

Ω u dx)p−1 −K and c = δ‖ψ‖−p∞ |Ω|1−p, we deduce that

y′(t) ≥ cyp(t) + g(t)∫

Ω

u dx, 0 < t < Tmax(u0). (43.2)

Now let u0 = µψ with µp−1 ≥ Kδ max

(1, |Ω|p−1

(∫Ω

ψ dx)1−p)

. On the one handwe have g(0) ≥ 0. On the other hand, we get∫

Ω

up0 dx− kup

0 = µp(∫

Ω

ψp dx− kψp)≥ µp(|Ω′| − k) = 2δµp ≥ Kµ ≥ −∆u0,

so that u ≥ u0 on (0, Tmax(u0)) by the comparison principle (Proposition 52.25).It follows that g(t) ≥ 0 on (0, Tmax(u0)) and (43.2) then implies Tmax(u0) < ∞.

To prove the existence of blowing-up solutions in the case p > q, we just notethat u satisfies

ut −∆u ≥∫

Ω

up dx− kup − Cu

for some 0 < k < |Ω| and C > 0. The result then follows by an obvious modificationof the proof in the case q = p.

(i2) Let us prove the local exponential stability of the trivial solution. It ob-viously suffices to treat the case k = 0. Let Θ be defined in (19.27), and putz(x, t) = ε(1 + Θ(x))e−αt. Then, for α, ε > 0 sufficiently small, we have

zt −∆z = ε(−α(1 + Θ) + 1)e−αt ≥ ε2e−αt

≥ εpe−pαt

∫Ω

(1 + Θ)p dx =∫

Ω

zp dx, t ≥ 0.

Therefore, if ‖u0‖∞ ≤ ε, then z is a supersolution to (43.1), so that u is globaland satisfies u(x, t) ≤ z(x, t) ≤ Ce−αt.

The local exponential stability of the trivial solution also follows from abstractresults in Appendix E (see Remark 51.20(ii)).

(ii) Multiplying the equation by um (m ≥ 1), integrating by parts and usingHolder’s inequality, we have

d

dt

∫Ω

um+1

m + 1dx + m

∫Ω

um−1|∇u|2 dx =∫

Ω

um dx

∫Ω

up dx− k

∫Ω

um+p dx

≤ (|Ω| − k)∫

Ω

um+p dx ≤ 0.

It follows from the Poincare inequality that

d

dt

∫Ω

um+1 dx ≤ − 4m

m + 1

∫Ω

|∇u(m+1)/2|2 dx ≤ −C

∫Ω

um+1 dx,

hence ∫Ω

um+1(t) dx ≤ M0e−Ct, 0 ≤ t < Tmax(u0).

If we choose m + 1 ≥ p, then (for different constants M0, C > 0) u satisfies

ut −∆u ≤ M0e−Ct, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (43.3)

Let now w(x, t) = M(1 + Θ(x))e−αt, with Θ defined in (19.27). If we chooseα ≤ min

((2(1 + ‖Θ‖∞))−1, C

)and M ≥ max(2M0, ‖u0‖∞), we then have

wt −∆w = (−α(1 + Θ) + 1)Me−αt ≥ M0e−Ct,

and w(0, .) ≥ u0, so that w is a supersolution of (43.3). It follows that u is globaland decays exponentially to 0 as t→∞.

(iii1) To show that all solutions of (43.1) are global and bounded, it suffices tonote that for any constant

M > max(‖u0‖∞, (|Ω|k−1)1/(q−p)

),

u ≡M is a supersolution of (43.1).(iii2) Next, let us show the global stability of 0 for k large. By Young’s inequality,

we have |Ω|up ≤ kuq + ε(k)u, where ε(k) = C(|Ω|, p, q)k−(p−1)/(q−p), so that usatisfies

ut −∆u ≤∫

Ω

up dx− |Ω|up + ε(k)u.

For k ≥ k0(|Ω|, p, q) sufficiently large (hence ε(k) small), the conclusion then fol-lows from an easy modification of the proof of assertion (ii).

(iii3) Finally, let us prove the existence of stationary solutions for k small. LetU = µΘ, with Θ again defined in (19.27) and µ = 2(

∫Ω Θp dx)−1/(p−1). For k > 0

sufficiently small we have

−∆U + kU q = (1 + kµq−1Θq)µ ≤ 2p−1µ =∫

Ω

Up dx.

By a modification of Proposition 52.20, it follows that the solution of (43.1) withu0 = U is nondecreasing in time, and we already know that it is global andbounded. Now Example 51.39 and Proposition 53.8 guarantee that u(t) → V inL∞(Ω) where V is a (classical) stationary solution, V ≥ U .

43.2. Blow-up rates, sets and profiles

In this subsection, we study the blow-up asymptotics for problem (43.1). Themethods and results of this subsection are from [480], except for Theorem 43.4 inthe case p ≥ 2 [487] and Theorem 43.11(ii) (which is an improvement of [487]).We refer to [60], [119], [525] for earlier results on blow-up asymptotics for problem(43.1) and its variants.

Our first result shows that blowing-up solutions to (43.1) exhibit global blow-upand can be described by a uniform blow-up profile in the interior of the domain.

Theorem 43.2. Assume Ω bounded, p > q ≥ 1, k ≥ 0, and 0 ≤ u0 ∈ L∞(Ω). Letu be the solution of (43.1) and assume that T := Tmax(u0) <∞. Then we have

limt→T

(T − t)1

p−1 u(x, t) = limt→T

(T − t)1

p−1 ‖u(t)‖∞ =[(p− 1)|Ω|

]− 1p−1 , (43.4)

uniformly on compact subsets of Ω.

Since the solution vanishes on the boundary and blows up globally inside Ω, itfollows that a boundary layer appears as t→ T . The following result describes thebehavior of the solution u near the blow-up time in the boundary layer.

Theorem 43.3. Under the assumptions of Theorem 43.2, for all K > 0, thereexist some constants C2 ≥ C1 > 0 and some t0 ∈ (0, T ), such that u satisfies

C1δ(x)√T − t

‖u(t)‖∞ ≤ u(x, t) ≤ C2δ(x)√T − t

‖u(t)‖∞, (43.5)

for all (x, t) in Ω× [t0, T ) such that δ(x) ≤ K√

T − t,

From the right-hand side of (43.5), one deduces that the size of the boundarylayer is at least of order

√T − t near the blow-up time, in the sense that u(x, t) =

o(‖u(t)‖∞), as t → T and δ(x)/√

T − t → 0. However, estimate (43.5) is notenough to conclude that the size of the boundary layer is exactly of order

√T − t,

in the sense that u(x, t)/‖u(t)‖∞ → 1, as t → T and δ(x)/√

T − t → ∞. Thefollowing theorem, though not very sharp regarding the actual behavior of thesolution in the boundary layer, enables one to conclude that this is indeed true.

Theorem 43.4. Under the assumptions of Theorem 43.2, for all ε > 0, thereexists C(ε) > 0 such that

u(x, t) ≥ ‖u(t)‖∞(1− ε− C(ε)

T − t

δ2(x)

), (x, t) ∈ Ω× [0, T ).

Therefore, we have

u(x, t)‖u(t)‖∞

−→

⎧⎪⎨⎪⎩1, as t→ T and δ(x)√

T − t→∞,

0, as t→ T and δ(x)√T − t

→ 0.

(43.6)

In other words, the size of the boundary layer decays like√

T − t.

Remarks 43.5. (a) Comparison with the local model problem. For prob-lem (15.1), we have seen that single-point blow-up occurs if Ω = BR and u ≥ 0 is ra-dial nonincreasing (cf. Theorem 24.1). If moreover p < pS, then u(·, t) behaves likeits maximum in space-time parabolas based at (0, T ), that is: u(x, t)/‖u(t)‖∞ → 1as t → T , uniformly for |x| ≤ C

√T − t. In some cases (see Remark 25.8), it is

even known that

u(x, t)‖u(t)‖∞

−→

⎧⎪⎪⎨⎪⎪⎩1, as t→ T and |x|√

(T − t)| log(T − t)|→ 0,

0, as t→ T and |x|√(T − t)| log(T − t)|

→ ∞.

(43.7)

At the opposite, blow-up for problem (43.1) is global and solutions behave like theirmaximum everywhere outside of a space-time parabolic neighborhood of (∂Ω, T )(compare formulas (43.6) and (43.7)). Problems (15.1) and (43.1) thus exhibit insome sense dual blow-up behaviors.

(b) Asymptotic influence of the local damping term. It appears fromTheorem 43.2 that the local damping term has no significant effect on the asymp-totic behavior of solutions near the blow-up time if q < p. In the blow-up criticalcase q = p, k < |Ω|, which was studied in [487], this is no longer so: Blow-up isstill global and uniform on compact sets, but the constant in the RHS of (43.4)has to be replaced by

[(p− 1)(|Ω| − k)

]−1/(p−1).

The proof of the above results relies on the study of linear problems with spa-tially homogeneous blowing-up source, of the form

ut −∆u = g(t), x ∈ Ω, 0 < t < T,

u = 0, x ∈ ∂Ω, 0 < t < T,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (43.8)

If g is a given function, locally Holder continuous on [0, T ), and if u0 ∈ L∞(Ω),then we know that (43.8) has a unique classical solution u ∈ C2,1(Ω× (0, T )), withu− e−tAu0 ∈ C(Ω× [0, T )).

In what follows we shall use the following notation. We set

G(t) =∫ t

0

g(s) ds and H(t) =∫ t

0

G(s) ds. (43.9)

We write u ∼ v for limt→T u(t)/v(t) = 1. As usual, λ1 and ϕ1 denote respec-tively the first Dirichlet eigenvalue and eigenfunction, normalized by

∫Ω ϕ1 dx = 1.

Moreover we setKρ = x ∈ Ω : δ(x) ≥ ρ, ρ > 0.

For problem (43.8), we shall prove the following theorem.

Theorem 43.6. Assume Ω bounded, 0 ≤ u0 ∈ L∞(Ω), g ≥ 0 locally Holdercontinuous on [0, T ). Let u ≥ 0 be the solution of (43.8). Then we have

lim supt→T

‖u(x, t)‖∞ =∞ (43.10)

if and only if ∫ T

0

g(s) ds =∞. (43.11)

Furthermore, if (43.10) or (43.11) is fulfilled, then

limt→T

u(x, t)G(t)

= limt→T

‖u(t)‖∞G(t)

= 1, (43.12)

uniformly on compact subsets of Ω.

The proof of Theorem 43.6 is based on eigenfunction arguments, one-sided esti-mates of ∆u (obtained via the maximum principle), and the mean-value inequalityfor subharmonic functions. We need the following two simple lemmas.

Lemma 43.7. (i) Under the assumptions of Theorem 43.6, we have

u ≤ G(t) + ‖u0‖∞, (x, t) ∈ QT . (43.13)

(ii) If moreover u0 ≡ 0, then

∆u ≤ 0, (x, t) ∈ QT . (43.14)

Proof. To prove (43.13) it suffices to notice that u(x, t) := G(t) + ‖u0‖∞ is asupersolution to (43.8).

To show (43.14) one could apply the maximum principle to the equation satisfiedby ∆u, after showing that ∆u ∈ C([0, T ), L2(Ω)). Alternatively, one can use thefollowing simple argument: For each h ∈ (0, T ), the function v(x, t) := u(x, t + h)−u(x, t) satisfies

vt −∆v = g(t + h)− g(t), x ∈ Ω, 0 < t < T − h,

v = 0, x ∈ ∂Ω, 0 < t < T − h,

v(x, 0) = u(x, h), x ∈ Ω.

⎫⎪⎬⎪⎭ (43.15)


Since u(·, h) ≤ G(h) due to (43.13), we see that v(x, t) := G(t + h) − G(t) is asupersolution to (43.15), hence u(x, t+h)−u(x, t) ≤ G(t+h)−G(t). Dividing byh and letting h → 0, we get ut ≤ g(t) in QT , hence (43.14).

As for the next lemma, a more accurate inequality will be given below to ob-tain precise boundary estimates, see (43.35). However this one is sufficient for thepurpose of Theorem 43.6.

Lemma 43.8. Assume Ω bounded and let z ∈ C2(Ω) satisfy

z ≥ 0 and ∆z ≥ 0, x ∈ Ω. (43.16)

Then

z(x) ≤ C(Ω)δn+1(x)

∫Ω

z(y)ϕ1(y) dy, x ∈ Ω.

Proof. Fix x ∈ Kρ ( = ∅). By the mean-value inequality for subharmonic func-tions, we have

z(x) ≤ 1|B(x, ρ/2)|

∫B(x,ρ/2)

z(y) dy =C(n)ρn

∫B(x,ρ/2)

z(y) dy.

Since infKρ ϕ1 ≥ c1(Ω)ρ and z ≥ 0, we deduce that

z(x) ≤ C(Ω)ρn+1

∫B(x,ρ/2)

z(y)ϕ1(y) dy ≤ C(Ω)ρn+1

∫Ω

z(y)ϕ1(y) dy

and the lemma follows.

Proof of Theorem 43.6. We first consider the case u0 ≡ 0. From (43.13), it isclear that (43.10) implies (43.11). Conversely, assume that (43.11) holds. Our aimis then to prove (43.12).

Define

z(x, t) = G(t)− u(x, t) and β(t) =∫

Ω

z(y, t)ϕ1(y) dy.

By Green’s formula, we have

β′(t) =∫

Ω

(g(t)− ut(y, t)

)ϕ1(y) dy = −

∫Ω

∆u(y, t)ϕ1(y) dy

= −∫

Ω

u(y, t)∆ϕ1(y) dy = λ1

∫Ω

u(y, t)ϕ1(y) dy = −λ1β(t) + λ1G(t).

Integrating this equation and using β(0) = 0, we obtain

β(t) = λ1

∫ t

0

eλ1(s−t)G(s) ds ≤ λ1H(t), (43.17)


where H is defined by (43.9). Since z ≥ 0 and ∆z ≥ 0 by (43.13) and (43.14),Lemma 43.8 implies

z(x, t) ≤ C(Ω)δn+1(x)

∫Ω

z(y, t)ϕ1(y) dy ≤ λ1C(Ω)H(t)ρn+1

, x ∈ Kρ, t ∈ (0, T ).

(43.18)For t close enough to T , we have G(t) > 0 by (43.11), and (43.13) and (43.18) giveus

0 ≤ 1− u(x, t)G(t)

≤ C(Ω)ρn+1

H(t)G(t)

, x ∈ Kρ. (43.19)

On the other hand, since G is nondecreasing, for all ε > 0 we have

0 ≤ H(t)G(t)

≤

∫ T−ε

0

G(s) ds,

G(t)+ ε.

Using (43.11), we deduce that limt→T H(t)/G(t) = 0. In view of (43.19), thisproves (43.12) for u0 ≡ 0.

Finally, for general u0 ≥ 0, we write u = U + e−tAu0, where U is the solution of(43.8) corresponding to u0 ≡ 0. By using ‖e−tAu0‖∞ ≤ ‖u0‖∞, the general caseeasily follows from the case u0 ≡ 0.

We are now in a position to prove Theorem 43.2.

Proof of Theorem 43.2. Case 1: k = 0. We apply Theorem 43.6 with

g(t) :=∫

Ω

up(y, t) dy, G(t) =∫ t

0

g(s) ds. (43.20)

By (43.12) in Theorem 43.6, it follows that

∀x ∈ Ω, limt→T

up(x, t)/Gp(t) = 1.

Moreover, (43.13) implies 0 ≤ up(x, t)/Gp(t) ≤ 2 in Ω for t close enough to T . ByLebesgue’s dominated convergence theorem, we infer that∫

Ω

up(y, t) dy ∼ |Ω|Gp(t), t→ T,

henceG′(t) = g(t) ∼ |Ω|Gp(t), (43.21)

or (G1−p)′ ∼ −(p− 1) |Ω|. After integrating this equivalence between t and T , weobtain

G(t) ∼[(p− 1)|Ω|(T − t)

]−1/(p−1). (43.22)


The result finally follows by returning to (43.12).Case 2: k > 0. It requires some modifications of the arguments from the case

k = 0 and from the proof of Theorem 43.6 (in particular we no longer consider thecase u0 ≡ 0 separately). We only indicate the necessary changes.

We first note that (43.13) and consequently (43.11) are still valid. As an analogueof (43.14) in Lemma 43.7, we next establish the inequality

∆u ≤ C1 := ‖∆u(·, T/2)‖∞, (x, t) ∈ Ω× [T/2, T ). (43.23)

By the strong maximum principle, we have u > 0 in QT . Set v = ∆u and notethat v ∈ C2,1(QT ) ∩ C(Ω × (0, T )) by parabolic regularity. Taking the Laplacianof equation (43.1) then yields

vt −∆v = −q(uq−1v + (q − 1)uq−2|∇u|2) ≤ −quq−1v in Ω× (0, T ),

with v(x, t) = −g(t) ≤ 0 on the boundary, where g is still defined by (43.20).Therefore, by the maximum principle, v cannot achieve an interior positive maxi-mum, hence (43.23).

Now set

z(x, t) = G(t) − u(x, t) + C1|x|22n

+ ‖u0‖∞ and β(t) =∫

Ω

z(y, t)ϕ1(y) dy.

By (43.13) and (43.23), we have

z ≥ 0 and ∆z ≥ 0, (x, t) ∈ Ω× (T/2, T ). (43.24)

On the other hand, arguing as in the proof of Theorem 43.6, we obtain

β′(t) = λ1

∫Ω

u(y, t)ϕ1(y) dy + k

∫Ω

uq(y, t)ϕ1(y) dy.

Integrating and using Holder’s inequality and (43.11), we get

β(t) ≤ C

(1 +

∫ t

0

∫Ω

uq(y, s) dy ds

)≤ C + C(T |Ω|)1−(q/p)

(∫ t

0

∫Ω

up(y, s) dy ds

)q/p

= o(G(t)),

(43.25)

as t→ T . Owing to (43.24) we may apply Lemma 43.8, and using (43.25) we thenconclude in a similar way as in the proof of Theorem 43.6 and Case 1.

To prove the boundary estimates in Theorems 43.3 and 43.4, we return toproblem (43.8) and introduce the following definition.


Definition 43.9. We say that g is sub-standard, resp. super-standard, if itsatisfies the following power-like growth assumption

g(t)/G(t) ≤ k1(T − t)−1, as t→ T , (43.26)

resp.g(t)/G(t) ≥ k2(T − t)−1, as t→ T , (43.27)

with constants k1, k2 > 0. We say that g is standard if it satisfies (43.26) and(43.27).

Note that if (43.26) holds, then g(t) ≤ C1(T − t)−(k1+1) as t → T . If (43.27)holds, then g(t) ≥ C2(T−t)−(k2+1) as t→ T , so that in particular

∫ T

0g(s) ds = ∞.

Conversely, g is standard whenever, for instance, c1(T − t)−α ≤ g(t) ≤ c2(T − t)−α

as t→ T , for some α > 1 and c2 ≥ c1 > 0.

Theorem 43.10. Assume Ω bounded, 0 ≤ u0 ∈ L∞(Ω), g ≥ 0 locally Holdercontinuous on [0, T ), and (43.11). Let u ≥ 0 be the solution of (43.8).(i) Assume that g is super-standard. Then for all K > 0 there exist C1 > 0 andt1 ∈ (0, T ), such that

u(x, t) ≥ C1δ(x)√T − t

G(t),


T − t.(ii) Assume that g is sub-standard. Then for all K > 0 there exist C2 > 0 andt2 ∈ (0, T ), such that

u(x, t) ≤ C2δ(x)√T − t

G(t),


T − t.

Theorem 43.11. Assume Ω bounded, 0 ≤ u0 ∈ L∞(Ω), g ≥ 0 locally Holdercontinuous on [0, T ). Let u ≥ 0 be the solution of (43.8).(i) Assume that G is super-standard. Then

u(x, t) ≥ ‖u(t)‖∞(1− C

T − t

δ2(x)

)− ‖u0‖∞ (43.28)

in Ω× [0, T ).(ii) Assume that g is super-standard and nondecreasing. Then for all ε > 0, thereexists C(ε) > 0 such that

u(x, t) ≥ ‖u(t)‖∞(1− ε− C(ε)

T − t

δ2(x)

)(43.29)

in Ω× [0, T ).

Note that estimate (43.28) is stronger than (43.29) when ‖u(t)‖∞ is large. How-ever, the assumption that G is super-standard is too restrictive in practice. For

instance in case of problem (43.1) with k = 0, we have g(t) :=∫Ω

up(t) dx ∼C(T − t)−p/(p−1) and G(t) ∼ C ′(T − t)−1/(p−1) by (43.21), (43.22), so that G issuper-standard only for p < 2, whereas g is standard for all p > 1.

For the proof of Theorem 43.10 we construct blowing-up sub-/supersolutionsof barrier type (relative to interior or exterior tangent balls at boundary points).As for Theorem 43.11, it is based on refinements of the arguments leading toTheorem 43.6.

Proof of Theorem 43.10. Step 1. We first claim that we need only considerthe case u0 ≡ 0. Indeed, for general u0, u may be decomposed as u = e−tAu0 + U ,where U solves Ut−∆U = g(t) with 0 initial and boundary values. Using e−tAu0 ∈C1,0(Ω×[T/2, T ]) and (43.11), we have 0 ≤ e−tAu0 ≤ C δ(x) ≤ ε δ(x)G(t)/

√T − t,

for all x ∈ Ω and t close enough to T . The claim follows.Step 2. We prove the lower estimate when u0 = 0. The basic idea is to seek a

suitable subsolution.Since Ω is smooth, ∂Ω satisfies a uniform interior and exterior sphere condition

i.e., for some R, R > 0 depending only on Ω, and for each point ξ ∈ ∂Ω, thereexist some balls Bi(ξ) of radius R and Be(ξ) of radius R such that Bi(ξ) ∩ Ωc =ξ = Be(ξ) ∩ Ω.

Now fix x0 ∈ Ω. Let ξ ∈ ∂Ω be such that δ(x0) = |x0− ξ|, and let B be the ballcontaining Bi(ξ), tangent to both Bi(ξ) and Be(ξ), of radius R = max(R, δ(x0)).By the definition of δ(x0), it is clear that B ⊂ Ω and that δ(x0) = dist(x0, ∂B),with R ≤ R ≤ diam(Ω). Without loss of generality, we may also assume that B iscentered at the origin. Define the space-time domain D = B × [0, T ), and divideD into two sub-regions as follows:

D1 : 0 ≤ d(x) <R

2√

T

√T − t, D2 : d(x) ≥ R

2√

T

√T − t, (43.30)

where d(x) = dist(x, ∂B) = R− r, r = |x|. We next define

v(x, t) =

⎧⎪⎪⎨⎪⎪⎩4G(t)

d(x)√T − t

(R√T− d(x)√

T − t

)in D1,

G(t)R2

Tin D2.

(43.31)

It is clear that v ∈ C1(D), and v(·, t) ∈ H2(B), 0 < t < T . Moreover, v(x, 0) = 0in B and v(x, t) = 0 for x ∈ ∂B. One then computes:

vt(x, t) =

⎧⎪⎪⎨⎪⎪⎩4d(x)√T − t

[G(t)T − t

(R

2√

T− d(x)√

T − t

)+ g(t)

(R√T− d(x)√

T − t

)]in D1,

g(t)R2

Tin D2.

(43.32)


We havevr(x, t) = vrr(x, t) = 0 in D2, (43.33)

while in D1 (where r ≥ R/2), we find that

vr(x, t) =4G(t)√T − t

(−R√

T+

2(R− r)√T − t

)and vrr(x, t) = −8G(t)

T−t , so that

−∆v(x, t) = −vrr−n− 1

rvr ≤

4G(t)T − t

(2+

n− 1r

R√T

√T − t

)≤ 8nG(t)

T − tin D1.

Therefore, we get

vt −∆v ≤

⎧⎪⎪⎨⎪⎪⎩G(t)T − t

R2

4T+ g(t)

R2

T+

8nG(t)T − t

in D1,

g(t)R2

Tin D2.

Using the fact that g is super-standard, it follows that vt −∆v ≤ C(R)g(t) in D,where C(R) = R2/T + (8n + R2/4T ) k−1

2 . Therefore, C(R)−1v is a subsolution inD, and since u ≥ 0, the maximum principle implies u ≥ C(R)−1v in D. On theother hand, for any K > 0, we have

v(x, t) ≥

⎧⎪⎪⎨⎪⎪⎩2R√

TG(t)

d(x)√T − t

, if d(x)/√

T − t ≤ R/2√

T ,

R2

TG(t) ≥ R2

TKG(t)

d(x)√T − t

, if R/2√

T ≤ d(x)/√

T − t ≤ K.

Since δ(x0) = d(x0), we deduce that if δ(x0) ≤ K√

T − t, then

u(x0, t) ≥ C1 G(t)δ(x0)√T − t

,

with

C1 =min(2R/

√T , R2/TK)

R2/T + (8n + R2/4T ) k−12

≥ C(T, K, n, k2)min(R, R2)

1 + R2

≥ C(T, K, n, k2)min(R2, diam−1(Ω)

),

where we have used R ≤ R ≤ diam(Ω). Therefore, C1 can be chosen independentof x0, and the desired lower estimate follows.

Step 3. We prove the upper estimate when u0 = 0. To do so, we show that thefunction v of Step 2, suitably modified and multiplied by a large constant, becomesa supersolution.


Fixing x0 ∈ Ω, and keeping the notation of Step 2, we now set D = B′c× [0, T ),with B′ = Be(ξ) the exterior ball, of radius R, associated with ξ, where ξ ∈ ∂Ωis such that δ(x0) = |x0 − ξ|. It is clear that δ(x0) = dist(x0, B

′) and we mayagain assume that B′ is centered at the origin. Consider the function v defined by(43.31), where now d(x) = dist(x, B′) = r − R and R = R/n, and where D1, D2

are still defined by (43.30). Formulae (43.32) and (43.33) are unchanged, whereasin D1 we now have

vr(x, t) =4G(t)√T − t

(R√T− 2(r −R)√

T − t

)

and vrr(x, t) = −8G(t)T−t , so that

−∆v(x, t) ≥ 4G(t)T − t

(2− n− 1

R

R√T

√T − t

)≥ 4G(t)

T − tin D1.

Therefore, we get

vt −∆v ≥

⎧⎪⎪⎨⎪⎪⎩4G(t)T − t

in D1,

R2

Tg(t) in D2.

Using the fact that g is sub-standard, we find that vt − ∆v ≥ C′(R)g(t) in D,where C′(R) = min(4k−1

1 , R2n−2T−1). It follows that C′(R)−1v is a supersolution

in D, hence in Ω × [0, T ), and the maximum principle implies u ≤ C ′(R)−1v, sothat

u(x0, t) ≤ C2 G(t)δ(x0)√T − t

in [0, T ),

with C2 = 4Rn−1T−1/2C′(R)−1, which proves the upper estimate.

Proof of Theorem 43.11. Step 1. We shall show that

u(x, t) ≥ G(t) − C(n)H(t)δ2(x)

, (x, t) ∈ Ω× [0, T ). (43.34)

In view of the maximum principle, it suffices to establish (43.34) for u0 ≡ 0, whichwe assume in the rest of this step. Estimate (43.34) is equivalent to the followinginequality, which is an improved version of Lemma 43.8:

supx∈Kρ

z(x, t) ≤ C(n)ρ2

H(t), (43.35)

where z(x, t) := G(t)− u(x, t). Note that z ≥ 0 due to (43.13).

We first establish (43.34) when Ω is a ball BR(x0). We may assume x0 = 0without loss of generality. Fix t ∈ (0, T ), x ∈ Ω and set ρ := R− |x|. Since ∆z ≥ 0by (43.14), the mean-value inequality for subharmonic functions implies

z(x, t) ≤ C(n)ρn

∫B(x,ρ/2)

z(y, t) dy. (43.36)

If ρ ≥ R/2, then

z(x, t) ≤ C(n)R1−n

ρ

∫Kρ/2

z(y, t) dy. (43.37)

Next suppose that ρ < R/2. Note that u(·, t) is radially symmetric due to u0 ≡ 0.Switching to polar coordinates, with z(y, t) = z(r, t), r = |y|, we may write

∫B(x,ρ/2)

z(y, t) dy =∫ |x|+ρ/2

|x|−ρ/2

z(r, t)M(r) dr,

where M(r) = Surf(B(x, ρ/2) ∩ S(0, r)) and “Surf” denotes the surface measure.Observing that

M(r) ≤ Surf(S(x, ρ/2)) ≤ C(n)ρn−1,

it follows from (43.36) that

z(x, t) ≤ C(n)ρ

∫ R−ρ/2

R/4

z(r, t) dr ≤ C(n)R1−n

ρ

∫ R−ρ/2

R/4

z(r, t)rn−1 dr,

so that (43.37) is true in all cases.Still assuming Ω = BR, fix ρ ∈ (0, R) and t ∈ (0, T ). Since the RHS in (43.37)

is a decreasing function of ρ and, for each x ∈ Kρ, ρ := R− |x| ≥ ρ, we see that

supx∈Kρ

z(x, t) ≤ C(n)R1−n

ρ

∫Kρ/2

z(y, t) dy. (43.38)

On the other hand, by (43.17) we have∫BR

z(y, t)ϕR(y) dy ≤ λRH(t), (43.39)

where λR is the first eigenvalue in BR and ϕR is the corresponding eigenfunction,normalized by

∫BR

ϕR = 1. By straightforward scaling arguments, we have

λR = C(n)R−2 and infKρ/2

ϕR ≥ c(n)R−(n+1)ρ. (43.40)

Inequality (43.35) then follows by combining (43.38), (43.39) and (43.40). There-fore (43.34) is proved when Ω = BR (and we stress that the constant C(n) doesnot depend on R).

To extend (43.34) to a general domain Ω, we fix x0 ∈ Ω and consider B =B(x0, R) ⊂ Ω with R = δ(x0). Letting u be the solution of ut − ∆u = g(t) inB× (0, T ), with 0 initial and boundary conditions, the maximum principle impliesu ≥ u. Since δ(x0) = dist(x0, ∂B), (43.34) follows from the same inequality in B.

Step 2. Let us show assertion (i) of the theorem. Since H(t) ≤ k−12 (T − t)G(t)

for t close to T by assumption, (43.34) and u ≥ 0 imply

u(x, t) ≥ G(t)(1− C

T − t

δ2(x)

)+, (x, t) ∈ Ω× [T0, T ), (43.41)

for some T0 ∈ (0, T ). By taking a larger constant C ≥ (T −T0)−1diam2(Ω), we seethat (43.41) becomes in fact valid in Ω× [0, T ). Estimate (43.28) then follows bycombining (43.41) and (43.13).

Step 3. To show assertion (ii) we shall use Step 1 to derive an estimate on ut

similar to (43.34), and then integrate over carefully chosen time intervals.Take u0 ≡ 0. Fix h > 0 and, for t ∈ [0, T − h), put v(·, t) = u(·, t + h) − u(·, t)

and g(t) = g(t+h)− g(t). Note that g ≥ 0 by assumption. The function v satisfies

vt −∆v = g(t), x ∈ Ω, 0 < t < T − h,

v = 0, x ∈ ∂Ω, 0 < t < T − h,

v(x, 0) = u(x, h), x ∈ Ω.

⎫⎪⎬⎪⎭ (43.42)

Applying the result of Step 1 to problem (43.42), we obtain

u(x, t + h)− u(x, t) ≥ G(t + h)−G(t)− C(n)H(t + h)−H(t)

δ2(x)

in Ω× [0, T − h). Dividing by h and letting h → 0, and next using the assumptionthat g is super-standard and u ≥ 0, we obtain

ut(x, t) ≥ g(t)−C(n)G(t)δ2(x)

≥ g(t)(1−C

T − t

δ2(x)

)+, (x, t) ∈ Ω×[T0, T ), (43.43)

for some T0 ∈ (0, T ). Fix γ > 1 and let Tγ := T − γ−1(T − T0). Let (x, t) ∈Ω × [Tγ , T ) and set tγ := T − γ(T − t) ∈ [T0, T ). Integrating (43.43) over (tγ , t)yields

u(x, t)−u(x, tγ) ≥(1−C

T − tγδ2(x)

)+

∫ t

tγ

g(s) ds =(1−γC

T − t

δ2(x)

)+

(G(t)−G(tγ)

).


Now, g being super-standard guarantees that s → (T − s)k2G(s) is nondecreasingfor s close to T . Taking Tγ closer to T , it follows that G(tγ) ≤ G(t)

(T−tT−tγ

)k2 =G(t)γ−k2 for t ∈ [Tγ , T ), hence

u(x, t) ≥ (1− γ−k2)G(t)(1− γC

T − t

δ2(x)

)+, (x, t) ∈ Ω× [Tγ , T ). (43.44)

By the maximum principle, (43.44) obviously remains true for u0 ≥ 0. Using(43.12), we get

u(x, t) ≥ ‖u(t)‖∞(1− 2γ−k2 − γC

T − t

δ2(x)

), (x, t) ∈ Ω× [Tγ , T ). (43.45)

Moreover, replacing γC by a larger constant C(γ), we see that (43.45) becomesin fact valid in Ω × [0, T ). Estimate (43.29) finally follows by choosing γ =(2/ε)1/k2 .

Proof of Theorems 43.3 and 43.4. Let g(t) = |Ω|− 1p−1[(p− 1)(T − t)

]− pp−1 . It

follows from Theorem 43.2 that, for all ε ∈ (0, 1), u satisfies (1−ε)g(t) ≤ ut−∆u ≤(1 + ε)g(t) in Ω× [Tε, T ) for Tε sufficiently close to T .

Taking, say, ε = 1/2, the maximum principle implies v ≤ u ≤ w in Ω× [T1/2, T ),where v and w solve vt −∆v = 1

2g(t) and wt −∆w = 32g(t) in Ω× [T1/2, T ) with

0 boundary values and initial conditions v(T1/2) = w(T1/2) = u(T1/2). Since gis standard, we deduce from Theorem 43.10 that v and w, hence u, satisfy theconclusion of Theorem 43.3.

Since g is standard and nondecreasing, by using the same comparison argu-ment (from below) for each ε ∈ (0, 1), along with Theorem 43.11(ii), we obtainTheorem 43.4.

Remark 43.12. Other nonlocal problems. We refer to e.g. [160], [331] forresults on systems of equations involving space integral terms. A different kind ofnonlocal equations, of “localized” type, have also been studied by several authors.A typical example is:

ut −∆u = up(x0(t), t), (43.46)

with Dirichlet boundary conditions. Here x0 : [0,∞) → Ω is a given (smooth)curve, which may be thought of as representing the location of a sensor drivingthe reaction in the whole domain. For equation (43.46), results on global (non-)existence can be found in [119], [479]. It is known that blow-up is global andthe asymptotics of blow-up was studied in [524], [480], [487] (the last two refer-ences contain results similar to Theorems 43.2–43.4). (Un-)boundedness of globalsolutions was investigated in [462], [488]. For other equations involving localizedterms, the blow-up set has been studied in [401], [221] (see Remark 44.4 below).Finally, results on systems of equations of localized type can be found in e.g. [411],[332].

43.3. Uniform bounds from Lq-estimates

In this subsection we derive smoothing estimates for problem (43.1), obtained in[463], which are similar to those obtained in Sections 15 and 16 for the model prob-lem (15.1). These estimates will be one of the main ingredients in the derivationof (universal) a priori bounds for global solutions in the next subsection. It turnsout that the critical value of q for smoothing from Lq into L∞ is smaller than forproblem (15.1) with the same p.

Theorem 43.13. Consider problem (43.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0,and 0 ≤ u0 ∈ L∞(Ω). Assume

q > qc :=np

n + 2.

There exists T = T (‖u0‖q) > 0 and C = C(Ω, p, q) > 0 such that Tmax(u0) > Tand

‖u(t)‖∞ ≤ C‖u0‖qt−n/2q, 0 < t < T.

Remarks 43.14. (a) The number qc in Theorem 43.13 is optimal (up to theequality case). Indeed, it was shown in [463] that for 1 ≤ q < qc (hence p > 1+2/n)there exists a sequence of nonnegative initial data u0,j ∈ L∞(Ω), bounded inLq, and such that Tmax(u0,j) → 0.

(b) On the other hand, it is not difficult to modify the arguments in the proofto show a local well-posedness result in Lq for q > qc, similar to Theorem 15.2.

Proof of Theorem 43.13. By the comparison principle (Proposition 52.25), itis sufficient to establish the result for k = 0. We proceed in two steps.

Step 1. We estimate the Lm-norm for m = max(p, q), by considering the quan-tity

H(t) := sups∈[0,t]

sα‖u(s)‖m, where α =n

2

(1q− 1

m

).

Using the variation-of-constants formula, ‖e−tAχΩ‖m ≤ C‖e−tAχΩ‖∞ ≤ C, m ≥ pand the Lq-Lm-estimate (cf. Proposition 48.4), we have

tα‖u(t)‖m ≤ tα‖e−tAu0‖m + tα∫ t

0

‖u(s)‖pp ‖e−(t−s)AχΩ‖m ds

≤ C‖u0‖q + Ctα∫ t

0

‖u(s)‖pm ds ≤ C‖u0‖q + CtαHp(t)

∫ t

0

s−pαds.

Since pα < 1 due to q > qc, by taking the supremum over (0, τ) we obtain

H(τ) ≤ C‖u0‖q + Cτ1−(p−1)αHp(τ), 0 < τ < Tmax(u0).

LetT := min

(1, ((2C)−p‖u0‖1−p

q )1/(1−(p−1)α)). (43.47)

We claim that

H(τ) ≤ 2C‖u0‖q, 0 < τ < min(T, Tmax(u0)). (43.48)

Indeed otherwise, since H(t) is continuous and H(0) = 0 (due to the regularity ofu), there exists a first τ < min(T, Tmax(u0)) such that

2C‖u0‖q = H(τ) ≤ C‖u0‖q + Cτ1−(p−1)α(2C‖u0‖q)p

hence τ ≥ T : a contradiction.Step 2. For 0 < t < min(T, Tmax(u0)), arguing as in Step 1 and using (43.47),

(43.48) and α ≤ n/2q, we get

tn/2q‖u(t)‖∞ ≤ tn/2q‖e−tAu0‖∞ + tn/2q

∫ t

0

‖u(s)‖pp ‖e−(t−s)AχΩ‖∞ ds

≤ C‖u0‖q + Ctn/2q

∫ t

0

‖u(s)‖pm ds

≤ C‖u0‖q + Ctn/2qHp(t)∫ t

0

s−pαds

≤ C‖u0‖q + CT 1−pα+n/2q‖u0‖pq ≤ C1‖u0‖q.

It follows in particular that Tmax(u0) > T and the theorem is proved.

43.4. Universal bounds for global solutions

In this subsection we prove universal bounds for global solutions of problem (43.1).It turns out that such bounds are true for all p > 1, in sharp contrast with themodel problem (15.1) (where even a priori estimates fail for p ≥ pS , cf. Theo-rem 28.7). The following result is due to [463].

Theorem 43.15. Consider problem (43.1) with Ω bounded, p > 1 and k = 0. Forall τ > 0, there exists C(Ω, p, τ) > 0 such that any global nonnegative solutionsatisfies

‖u(t)‖∞ ≤ C(Ω, p, τ), t ≥ τ. (43.49)

As an important ingredient of the proof, we first establish uniform a prioriestimates for global solutions. Note that the problem does not seem to admit anenergy functional and that the proof, based on maximum principle arguments, iscompletely different from that of Theorem 22.1.

Proposition 43.16. Consider problem (43.1) with Ω bounded, p > 1 and k = 0.For all M > 0, there exists K(Ω, p, M) > 0 such that any global nonnegativesolution with ‖u0‖∞ ≤M satisfies

‖u(t)‖∞ ≤ K, t ≥ 0. (43.50)

Proof. In this proof, we denote g(t) :=∫Ω up(t) dx and assume that ‖u0‖∞ ≤M .

Step 1. We first establish a (universal) integral bound on the source term:∫ t+1

t

g(s) ds ≤ C(Ω, p), t ≥ 0. (43.51)

We argue as in the proof of Theorem 17.1 and denote y = y(t) :=∫Ω u(t)ϕ1 dx.

Multiplying the equation with ϕ1, integrating by parts and using∫Ω ϕ1 dx = 1, we

obtainy′ + λ1y =

∫Ω

up dx. (43.52)

By Holder’s inequality, we deduce that

y′ ≥ −λ1y + C1yp

with C1 = ‖ϕ1‖−p∞ |Ω|1−p. It follows that y(t) ≤ C2 := (λ1/C1)1/(p−1) for all t ≥ 0,since otherwise u cannot exist globally. Integrating (43.52) in time, we deduce(43.51) with C = (1 + λ1)C2.

Step 2. This is the main step: We shall show that u becomes eventually monotoneif g(t) reaches a suitably large value.

Comparison with the solution of the ODE y′ = |Ω|yp, y(0) = M , shows thatthere exists t0 = t0(M) > 0 such that

‖u(t)‖∞ ≤ 2M, 0 < t ≤ t0. (43.53)

Now Lp- and Schauder estimates guarantee that there exists K1 = K1(M) > 0such that

‖∆u(t0)‖∞ ≤ K1. (43.54)

We claim that:

if g(t1) ≥ K1 for some t1 ≥ t0, then ut ≥ 0 in Ω× [t1,∞). (43.55)

Thus assume t1 ≥ t0 and g(t1) ≥ K1, and pick t2 ∈ [t0, t1] such that

g(t2) = max[t0,t1]

g(t) ≥ K1. (43.56)


Let v := ∆u and w =: ut. By parabolic regularity results, we have v, w ∈C2,1(QT ) ∩C(Ω× (0, T )). Since v satisfies

vt −∆v = 0, x ∈ Ω, t > t0,

v = −g(t), x ∈ ∂Ω, t > t0,

v(x, t0) = ∆u(x, t0), x ∈ Ω,

⎫⎪⎬⎪⎭we deduce from the maximum principle, (43.54) and (43.56) that

∆u ≥ min(min

Ω∆u(·, t0),−g(t2)

)= −g(t2), x ∈ Ω, t ∈ [t0, t1].

Consequently,ut(·, t2) = ∆u(·, t2) + g(t2) ≥ 0.

Since w satisfies

wt −∆w = p

∫Ω

up−1w dy, x ∈ Ω, t > t2,

w = 0, x ∈ ∂Ω, t > t2,

w(x, t2) ≥ 0, x ∈ Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭where up−1 ≥ 0, we deduce from the maximum principle for nonlocal equations(see Proposition 52.24) that ut ≥ 0 in Ω× [t2,∞), which implies the claim.

Step 3. We next deduce a uniform estimate on the source term: there existsK2 = K2(M) > 0 such that

g(t) ≤ K2, t ≥ 0. (43.57)

Indeed, if g(t1) ≥ K1 for some t1 ≥ t0, then

g(t) ≤∫ t+1

t

g(s) ds ≤ C(Ω, p), t ≥ t1

by (43.55) and (43.51). Consequently, taking also (43.53) into account, we get(43.57) with K2 := max(K1, C(Ω, p), |Ω|(2M)p).

Step 4. Conclusion. Let Θ be defined in (19.27). Owing to (43.57), we see thatu := ‖u0‖∞ + K2Θ is a supersolution to (43.1). Consequently, (43.50) with K =‖u0‖∞ + K2‖Θ‖∞ follows from the comparison principle (Proposition 52.25).

Proof of Theorem 43.15. By (43.51), there exists t0 ∈ (0, τ/2) such that

‖u(t0)‖p ≤ C(Ω, p)τ−1/p.

Since p > qc = np/(n + 2), applying Theorem 43.13, we infer the existence oft1 ∈ (t0, τ) such that

‖u(t1)‖∞ ≤ C(Ω, p, τ). (43.58)

Estimate (43.49) finally follows by combining (43.58) and (43.50) (taking t1 asinitial time).

44. Problems involving space integrals (II)

In this section, we consider a different class of nonlocal equations, of the form

ut −∆u =(∫

Ω

g(u)dx)m

f(u), (44.1)

with Ω bounded and m ∈ R, m = 0.

44.1. Transition from single-point to global blow-up

We have seen in the previous section that purely nonlocal power nonlinearities giverise to global blow-up with a uniform profile (for all nonglobal solutions), whereaspurely local power nonlinearities produce single-point blow-up in the radial non-increasing case (cf. Theorem 24.1). In order to understand the transition betweenthese two complementary situations, it is natural to consider equation (44.1) withf(u) = uq, g(u) = up−q, p > 1, 0 < q 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (44.2)

In what follows we assume u0 ∈ L∞(Ω), u0 ≥ 0, u0 ≡ 0, and we shall denote

g(t) =∫

Ω

up−q(y, t) dy, G(t) :=∫ t

0

g(s) ds.

Remark 44.1. Non-Lipschitz case. Problem (44.2) is well-posed for q ≥ 1 andp ≥ q+1. In the non-Lipschitz cases 0 < q < 1 and/or 0 < p−q < 1, existence of alocal classical solution can still be shown, either by using the Schauder fixed pointtheorem, or by an approximation procedure (replacing the initial and boundaryconditions by uε(x, 0) = u0(x) + ε and uε(x, t) = ε, respectively). However, localuniqueness seems to be unknown in this case, and assertion (i) of Theorem 44.2applies to any maximal solution starting from u0. By a simple modification of theproof of Theorem 17.1, any solution u starting from suitably large u0 will blow upin a finite time T = T (u), in the sense that lim supt→T ‖u(t)‖∞ = ∞.

The following result shows that the occurrence of single-point vs. (uniform ornonuniform) global blow-up depends in a precise way on the values of q. Observethat the rate of the (uniform) global blow-up does not change when q varies in[0, 1) and that the bifurcation to single-point for q in (1, p], occurs through anonuniform global blow-up at q = 1. Theorem 44.2 is a variant of results combinedfrom [157] and [331], except for the blow-up rate estimates (44.6)–(44.7) which areconsequences of Proposition 44.3 below.

44. Problems involving space integrals (II) 399

Theorem 44.2. Assume Ω bounded, p > 1, 0 < q < p, and 0 ≤ u0 ∈ L∞(Ω). Letu be a nonglobal solution of problem (44.2) and denote by T its maximal existencetime.(i) If 0 < q < 1, then blow-up is global and uniform. More precisely:

limt→T

(T − t)1

p−1 u(x, t) = limt→T

(T − t)1

p−1 ‖u(t)‖∞ =[(p− 1)|Ω|

]− 1p−1 , (44.3)

uniformly on compact subsets of Ω.(ii) If q = 1, then blow-up is global and nonuniform. More precisely:

u(x, t) = k(t)(e−tAu0

)(x), where k(t) ∼ C(T − t)−1/(p−1) as t→ T (44.4)

for some constant C > 0 depending on u0.(iii) Assume 1 < q < p, Ω = BR, u0 ∈ C1(Ω) radial nonincreasing, with u0(x) = 0for |x| = R. Then single-point blow-up occurs at x = 0. More precisely, for anyα > 2/(q − 1) there exists Cα > 0 such that

u(x, t) ≤ Cα|x|−α, 0 < |x| < R, 0 < t < T. (44.5)

Assume in addition that p− q < n(q − 1)/2. Then we have

‖u(t)‖∞ ≥ C1(T − t)−1/(p−1), 0 < t < T (44.6)

and, if in addition q < pS, then

‖u(t)‖∞ ≤ C2(T − t)−1/(p−1), 0 < t < T, (44.7)

for some C1, C2 > 0.

Proof. (i) Set v = 11−qu1−q and let

M(t) := maxΩ

u(·, t), N(t) := maxΩ

v(·, t).

By the argument in the (alternative) proof of Proposition 23.1, we have M ′(t) ≤M q(t)g(t), hence N ′(t) ≤ g(t) a.e. in (0, T ). Consequently,

N(t) ≤ N(0) + G(t), 0 < t < T, (44.8)

hence in particularlimt→T

G(t) =∞. (44.9)

On the other hand, noting that u > 0 in QT by the strong maximum principle,we have

vt −∆v = u−q(ut −∆u) + qu−q−1|∇u|2 ≥ g(t).

By using (44.9), Theorem 43.6 and the maximum principle, it follows that, uni-formly on compact subsets, lim inft→T v(x, t)/G(t) ≥ 1, hence

limt→T

v(x, t)G(t)

= 1 (44.10)

by (44.8). Arguing as in the proof of Theorem 43.2 for k = 0, we obtain after somecalculations

G(t) ∼ (1 − q)−1[(p− 1)|Ω|(T − t)

]−(1−q)/(p−1).

Returning to (44.10), (44.8) and using u = ((1 − q)v)1/(1−q) we obtain (44.3).

(ii) For q = 1, by direct calculation one checks that the solution of (44.2) canbe written as

u(x, t) = eG(t)e−tAu0,

and we have G(t) →∞ as t→ T . Consequently,

g(t) = e(p−1)G(t)

∫Ω

(e−tAu0

)p−1dx

hence

d

dte−(p−1)G(t) = −(p−1)g(t)e−(p−1)G(t) → −C, C := (p−1)

∫Ω

(e−TAu0)p−1 dx,

as t → T . By integration, we obtain eG(t) ∼ C1/(p−1)(T − t)−1/(p−1) and (44.4)follows.

(iii) The proof of (44.5) is very similar to that of Theorem 24.1. The variablesf , f ′ now stand for f = f(t, u) = g(t)uq, f ′ = g(t)quq−1, and J is defined by(24.3) with 1 < γ < q. The main difference is that the condition H ≥ 0 becomesequivalent to

g(t)(q − γ)uq−1 + (n + δ)δr−2 ≥ 2εγ(1 + δ)uγ−1rδ, (44.11)

instead of (24.5). Since

g(t) ≥∫

Ω

(e−tAu0

)p−qdx ≥ c > 0, 0 ≤ t < T, (44.12)

(44.11) is satisfied if ε is small enough.

To show the blow-up estimates (44.6)–(44.7) in Theorem 44.2(iii), we first es-tablish the following more general result, where the upper bound will be provedby using arguments from [425] (cf. Theorem 26.8).

Proposition 44.3. Let T > 0, p > 1, and let a ∈ C([0, T )) be nonnegative andbounded. Let 0 ≤ u ∈ C2,1(BR × (0, T )) be a radial nonincreasing solution of theequation

ut −∆u = a(t)up, x ∈ BR, 0 < t < T,

such that limt→T ‖u(t)‖∞ = ∞.(i) There exists C1 > 0 such that

‖u(t)‖∞ ≥ C1(T − t)−1/(p−1), 0 < t < T.

(ii) Assume in addition that := limt→T a(t) exists in (0,∞) and that p < pS.Then there exists C2 > 0 such that

‖u(t)‖∞ ≤ C2(T − t)−1/(p−1), 0 < t < T. (44.13)

Proof. (i) Since N(t) := sup|x|<R u(x, t) = u(0, t), we have ur(0, t) = 0 andurr(0, t) ≤ 0. Therefore, dN/dt ≤ a(t)Np(t) and assertion (i) follows immediatelyupon integration.

(ii) We shall apply Lemma 26.11 with D = (0, T ), Σ = X = [0, T ] and thestandard distance. We denote d(s) = dist(s, ∂D) = min(s, T − s). Assume thatestimate (44.13) fails and denote M(t) = up−1(0, t). Then there exists a sequencesk → T such that

M(sk) > 2k(T − sk)−1 = 2kd−1(sk). (44.14)

It follows from Lemma 26.11 that there exists tk ∈ (0, T ) such that

M(tk)d(tk) > 2k, (44.15)

M(tk) ≥M(sk) (44.16)

and

M(t) ≤ 2M(tk) for all t ∈ (0, T ) ∩ (tk − kM−1(tk), tk + kM−1(tk)). (44.17)

Note that, by (44.14) and (44.16) we have

tk → T. (44.18)

For k large, we deduce from (44.15) that kM−1(tk) < d(tk) = T − tk, so that(44.17) rewrites as

M(t) ≤ 2M(tk) for all t ∈ (tk − kM−1(tk), tk + kM−1(tk)). (44.19)

Now we rescale uk by setting

λk := M−1(tk)→ 0 (44.20)

and

vk(y, s) := λ1/(p−1)k uk(λ1/2

k y, tk + λks), (y, s) ∈ Dk := |y| < Rλ−1/2k × (−k, k).

The function vk solves

∂svk −∆yvk = a(tk + λks)vpk, (y, s) ∈ Dk. (44.21)

Moreover we have vk(0, 0) = 1 and (44.19) implies

0 ≤ vk ≤ C := 21/(p−1), (y, s) ∈ Dk. (44.22)

By using (44.21), (44.22), (44.20), (44.18), interior parabolic estimates and theembedding (1.2), we deduce that some subsequence of vk converges in Cα(Rn+1),0 < α < 1, to a (bounded classical) solution v ≥ 0 of

vt −∆v = vp, x ∈ Rn, s ∈ R.

Moreover, v is radial nonincreasing and satisfies v(0, 0) = 1. This contradictsTheorem 21.1.

End of proof of Theorem 44.2. In view of Proposition 44.3, to show (44.6)and (44.7), it suffices to verify that

g(t)→ ∈ (0,∞), as t→ T . (44.23)

Due to (44.5) and p − q < n(q − 1)/2, the function g(t) is bounded on [0, T ).By (44.5), parabolic estimates and the embedding (1.2), it follows that, for someν ∈ (0, 1), u ∈ BUCν(γ < |x| < 1− γ× (T/2, T )) for each γ > 0. Consequently,for all x ∈ B(0, 1)\0, limt→T u(x, t) exists and is finite. Using (44.5) and p−q <n(q − 1)/2 again, along with the dominated convergence theorem and (44.12), weobtain (44.23).

Remark 44.4. Problems involving localized nonlinearities. A different typeof competition between local and nonlocal reaction terms has been studied in [401],[221] for the following variant of equation (43.46):

ut −∆u = uq(x0, t) + up

p > 1, q > 0, x0 ∈ Ω, with Dirichlet boundary conditions, when Ω is a ball BR andu0 is radial decreasing. Interestingly, the critical condition is different dependingon the location of x0. Namely, for x0 = 0, blow-up is always global if p ≤ q + 1,while single-point blow-up occurs for some u0 if p > q + 1. Next assume x0 = 0.If p < q, then both global and single-point blow-ups occur, and there are no otherpossibilities. On the contrary, if p > q (or p = q > 2), then only single-pointblow-up occurs.

44.2. A problem with control of mass

We now consider equation (44.1) with f(u) = g(u) = up, p > 1, and m = −1,under Neumann boundary conditions, that is:

ut −∆u =(∫

Ω

up(y, t) dy)−1

up, x ∈ Ω, t > 0,

uν = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (44.24)

In what follows we assume u0 ∈ L∞(Ω), u0 ≥ 0, u0 ≡ 0, and we shall denote

k(t) =(∫

Ω

up(y, t) dy)−1

.

As in Chapters I and II we shall use the notation

psg = ∞ if n ≤ 2,

n/(n− 2) if n > 2.

Let us first observe that, by integrating the equation, we immediately obtain∫Ω

u(t) dx = t +∫

Ω

u0 dx. (44.25)

This means that the total “mass” is controlled.

We shall first investigate under what conditions the solutions of (44.24) blow upor exist globally. On a heuristic level one can expect that, when u becomes largein some sense, then the factor k(t) might become large, too, and have a stabilizingeffect which could prevent blow-up. Interestingly, whether or not this possiblestabilizing effect is effective depends in a sharp way on the relation between theexponent p and the space dimension n. The following result is due to [284].

Theorem 44.5. Consider problem (44.24) with Ω bounded, p > 1, and 0 ≤ u0 ∈L∞(Ω), u0 ≡ 0.

(i) If p < psg, then Tmax(u0) = ∞ for all u0.

(ii) Assume n ≥ 3, p > psg and let Ω = B1. Then there exists u0 such thatTmax(u0) <∞.

Note that since the solution stays bounded in L1, it is clear that for radialnonincreasing solutions, blow-up can occur only at the origin. As a corollary tothe proof of Theorem 44.5, one obtains the following blow-up profile estimate.

Theorem 44.6. Consider problem (44.24) with n ≥ 3, p > psg and Ω = B1.Then there exists 0 ≤ u0 ∈ L∞(Ω), u0 ≡ 0, radial nonincreasing, such that T :=Tmax(u0) <∞, u exhibits single-point blow-up at x = 0, and u satisfies

u(x, t) ≤ Cε|x|−2

p−1−ε, x ∈ Ω, 0 < t < T, for each ε > 0. (44.26)

Moreover, (44.26) is optimal, in the sense that it cannot be satisfied for any ε < 0.

As for the blow-up rate, we have the following result, which is a consequence ofProposition 44.3.

Theorem 44.7. Consider problem (44.24) with p > 1 and Ω = B1. Let 0 ≤ u0 ∈L∞(Ω), u0 ≡ 0, be radial nonincreasing and assume that T := Tmax(u0) < ∞.(i) There exists C1 > 0 such that

‖u(t)‖∞ ≥ C1(T − t)−1/(p−1), 0 < t < T.

(ii) Assume in addition that p < pS and that u satisfies (44.26). Then there existsC2 > 0 such that

‖u(t)‖∞ ≤ C2(T − t)−1/(p−1), 0 < t < T. (44.27)

Remarks 44.8. (a) Global solutions. For any p > 1, (44.24) admits globalsolutions for arbitrarily large initial data. Namely it suffices to take homogeneousinitial data u0 = M (with any M > 0) and u is then given by solving the ODE, i.e.:uM (x, t) = M + |Ω|−1t. We thus observe that it is the “shape” of u0, rather thanits size, which causes blow-up. On the other hand, all global solutions of (44.24)are unbounded, due to (44.25).

(b) Failure of the comparison principle. Problem (44.24) admits no com-parison principle. For instance taking Ω = B1 and u a blow-up solution as inTheorem 44.6 (ii), we see that u(·, 0) < M for M large but u eventually intersectsthe solution uM (x, t) = M + |Ω|−1t.

(c) Interpretation of the critical exponent. Observe that k(t) is boundeddue to (44.25) and Holder’s inequality and that k(t) vanishes if and only if ‖u(t)‖p

blows up. This allows for a heuristic interpretation of the value of the critical expo-nent in Theorem 44.5 if we put problem (44.24) in parallel with the model equationut − ∆u = up. Indeed the supercriticality condition for the Lp-norm is given byp > n(p − 1)/2 that is, p < psg. More precisely, for the model problem (15.1),if p < psg, then ‖u(t)‖p blows up whenever u is nonglobal (cf. Theorem 15.2),whereas if p > psg, then there exist solutions (in a ball) such that ‖u(t)‖p remainsbounded (cf. Theorem 24.1 and Corollary 24.2). The idea of the proof of Theo-rem 44.5(ii) below is precisely to use (a nontrivial modification of) the method in

Theorem 24.1 to construct initial data which yield a blow-up profile belonging toLp and provide a control of k(t) from below.

(d) Critical case. In the critical case p = psg, it is proved in [284] that thesolution exists globally if

∫Ω

u0 dx is large enough.

(e) Explicit examples of initial data in Theorem 44.5(ii) are constructed inLemma 44.10 below. Namely blow-up occurs whenever u0 ∈ C2(B1) is radial andsatisfies (44.33)–(44.37) with β > 0 small (depending on n, p) and M > 0 large(depending on n, p, β).

Remarks 44.9. Other nonlocal problems. Results on blow-up for other non-local problems with control of mass, of the form ut −∆u = f(u)− 1

|Ω|∫Ω

f(u) dy

with Neumann boundary conditions and f(u) = |u|p or |u|p−1u, can be found in[103], [284]. For physical motivation concerning such problems, see [465].

Proof of Theorem 44.5(i). The proof here is for n ≥ 3. The cases n = 1, 2 canbe obtained with obvious modifications. Fix m > 1. For any 0 < a < 1 < q, wehave ∫

Ω

up+m dx ≤(∫

Ω

u(p+m)aq dx)1/q(∫

Ω

u(p+m)(1−a)q′dx)1/q′

.

We claim that we can find 0 < a < 1 and q > n/(n− 2) such that

(p + m)aq = (m + 1)n/(n− 2) and (p + m)(1− a)q′ ≤ p. (44.28)

Indeed, (44.28) is equivalent to

a =n

(n− 2)qm + 1m + p

≥ 1− p

m + p

(1− 1

q

),

i.e.:

q ≤ n

n− 2+

1m

( n

n− 2− p)

(44.29)

and, since p < n/(n− 2), we can choose q > n/(n− 2) satisfying (44.29) and thecorresponding a then belongs to (0, 1).

Now using Holder’s inequality, we obtain∫Ω

up+m dx ≤ C(∫

Ω

u(m+1)n/(n−2) dx)1/q(∫

Ω

up dx)(p+m)(1−a)/p

. (44.30)

Multiplying (44.24) by um and integrating by parts over Ω, we obtain

d

dt

∫Ω

um+1

m + 1dx + m

∫Ω

um−1|∇u|2 dx =(∫

Ω

up dx)−1

∫Ω

up+m dx. (44.31)

Set v = u(m+1)/2. Since∫Ω

up dx ≥ C(∫

Ω

u dx)p

≥(∫

Ω

u0 dx)p

(44.32)

by (44.25), formulas (44.30), (44.31) and (p + m)(1− a) n/(n− 2),we obtain

d

dt

∫Ω

v2 dx ≤ C(1 +

∫Ω

v2 dx).

By integration, it follows that for all m > 1, τ > 0,∫Ω

um+1(t) dx =∫

Ω

v2(t) dx ≤ C(m, τ), 0 < t < min(τ, T ).

Therefore, using also (44.32), the right-hand side of (44.24) remains bounded in Lr

on bounded time intervals for each r <∞. The L∞-boundedness of u on boundedtime intervals then follows easily from the variation-of-constants formula and theLp-Lq-estimates (cf. Proposition 48.4). We conclude that u exists globally.

The proof of part (ii) is more delicate. It requires carefully constructed initialdata. This is achieved in the following lemma.

Lemma 44.10. Let Ω = B1 and p > psg. Then, for all M, β > 0, one can find aradial function u0 ∈ C2(Ω) satisfying the following properties:

u0(0) ≥ M, u0(1) = β, u0,r(1) = 0, u0,r < 0 on (0, 1), (44.33)∫Ω

u0 dx ≤ Cβ, (44.34)

k(0) =(∫

Ω

up0 dy

)−1

≥ Aβ−p, (44.35)

∆u0 + λup0 ≥ 0, |x| ≤ 1, (44.36)

u0,r + µrup0 ≤ 0, 0 ≤ r ≤ 1/2, (44.37)

where λ = Kβ1−p, µ = Lβ1−p, and C, A, K, L > 0 depend only on n, p.

Proof. Let α = 2/(p− 1) and fix a function U ∈ C2((0, 1]) such that

U(r) = r−α on (0, 1/2], Ur < 0 on (0, 1), Ur(1) = 0 and U(1) = 1.(44.38)

Fix δ ∈ (0, 1/4) and β > 0. We define

φ(r) :=

U(r), δ < r ≤ 1,

δ−α(1 + α(α+5)

6 − α(α+3)2 ( r

δ )2 + α(α+2)3 ( r

δ )3), 0 ≤ r ≤ δ,

(44.39)

and we set u0 = βφ. One can check that u0 ∈ C2(Ω), that 0 ≤ u0 ≤ βU on (0, 1],and that u0 satisfies (44.33) whenever 0 < δ ≤ (M/β)−1/α.

Since pα < n, we have∫Ω Up dx < ∞, hence (44.34) and (44.35). On the other

hand, we have

∆φ + Kφp ≥

⎧⎪⎪⎨⎪⎪⎩∆U + K, 1/2 ≤ |x| ≤ 1,(α(α + 2− n) + K

)r−α−2, δ ≤ |x| ≤ 1/2,(

−nα(α + 3) + K)δ−α−2, |x| ≤ δ.

Since ∆u0 + Kβ1−pup0 = β(∆φ + Kφp) this implies (44.36) for K = K(n, p) > 0

large.Next we have

φr + Lrφp ≤ −αr−α−1 + Lr−αp+1 = (L − α)r−α−1, δ ≤ r ≤ 1/2,

and

φr + Lrφp ≤ δ−α(−α(α+3)r

δ2 + α(α+2)r2

δ3

)+ LC(α)δ−pαr ≤ δ−αp(LC(α) − α)r

for 0 ≤ r ≤ δ. Since u0,r + Lβ1−prup0 = β(φr + Lrφp), this implies (44.37) for

L = L(p) > 0 small.

Next, we consider the auxiliary problem

wt −∆w = 2λwp, x ∈ Ω, t > 0,

uν = 0, x ∈ ∂Ω, t > 0,

w(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (44.40)

We shall need the following upper estimate on the existence time of its solution.

Lemma 44.11. Let Ω = B1 and p > psg. For M, β > 0, let λ and u0 be asin Lemma 44.10. Then the existence time Tw of the solution of problem (44.40)satisfies Tw ≤ M1−p

λ(p−1) .

Proof. We use a similar idea as in the proof of Theorem 23.5, applying the maxi-mum principle to the auxiliary function J := wt−λwp. By the maximum principlewe have w ≥ β > 0 in Q := Ω× (0, Tw). On the other hand, Example 51.9 shows

that w ∈ C([0, Tw), W 1,q(Ω)) for any q > n, hence wp ∈ C([0, Tw), W 1,2(Ω)) andTheorem 51.1(v) guarantees wt ∈ C([0, Tw), L2(Ω)). In addition, wt ∈ C2,1(Ω ×(0, Tw)) (cf. Example 51.10). Therefore, J ∈ C([0, Tw), L2(Ω))∩C2,1(Ω× (0, Tw)).

Now, J satisfies

Jt −∆J = (wt −∆w)t − λp(wp−1wt − wp−1∆w − (p− 1)wp−2|∇w|2

)≥ 2λpwp−1wt − λpwp−12λwp = 2λpwp−1J

in Q. At t = 0, we have J = ∆u0 + λup0 ≥ 0 by (44.36). On ∂Ω, we have

∂J∂ν = ∂

∂t (∂w∂ν )−λpwp−1 ∂w

∂ν = 0. It thus follows from the maximum principle (cf. Re-mark 52.9) that J ≥ 0 in Ω× (0, Tw). But this implies (w1−p)t ≤ −λ(p− 1), hencein particular w1−p(0, t) + λ(p− 1)t ≤ u1−p

0 (0) ≤ M1−p on (0, Tw) by (44.33). Thelemma follows.

Observe that, for β ≤ A/4K, we have k(0) ≥ Aβ−p ≥ 4Kβ1−p = 4λ. The ideaof the proof is now to show that k(t) cannot become smaller than 2λ before thetime t = β, independently of M . This will be achieved via the next lemma, whereu is estimated from above by employing a modification of an argument from [219](cf. Theorem 24.1). This will guarantee that u dominates the solution w of theauxiliary problem (44.40) for t ≤ β. But the blow-up time of w goes to 0 as Mincreases, which will imply blow-up of u if M is large.

Lemma 44.12. Let Ω = B1 and p > psg. For M > 0 and β ∈ (0, 1), let A, K, λand u0 be as in Lemma 44.10. Set T0 = min(β, T ). Assume in addition that β ≤A/4K, so that k(0) ≥ 4λ, and define

T1 = supτ ∈ [0, T0) : k(t) ≥ 2λ on [0, τ ]

∈ (0, T0].

For each 1 < q < p, we have

u(r, t) ≤ C(n, p, q)βr−2/(q−1), 0 < r ≤ 1, 0 < t < T1. (44.41)

Proof. Step 1. By Example 51.13, we have ur ∈ C2,1((0, 1)× (0, T )) ∩ C([0, 1]×[0, T )). Using (44.33) and the maximum principle (in particular Proposition 52.17),one deduces that

u ≥ β and ur ≤ 0, 0 ≤ r ≤ 1, 0 < t < T. (44.42)

Since ur ≤ 0, we have

u(r, t) ≤ nr−n

∫ r

0

u(ρ, t)ρn−1dρ ≤ C(n)r−n

∫Ω

u(t) dx.

Using (44.25), (44.34) and T0 ≤ β, we deduce that

u(r, t) ≤ C(n, p)βr−n, 0 < r ≤ 1, 0 < t < T0. (44.43)

We next claim that

ur(1/2, t) ≤ −c(n, p)β, 0 ≤ t < T0. (44.44)

To show this, observe that the function v := β−1ur satisfies

vt − vrr +n− 1

r2v − n− 1

rvr = pk(t)up−1v ≤ 0, 1/4 < r < 1, 0 < t < T,

v(1/4, t) ≤ 0, v(1, t) = 0, 0 < t < T,

v(0, r) = Ur(r) < 0, 1/4 < r < 1,

⎫⎪⎪⎬⎪⎪⎭where U , defined in (44.38), depends only on p for r ∈ (1/4, 1). By the strongmaximum principle, recalling that T0 ≤ β < 1, it follows in particular thatv(1/2, t) ≤ −c(n, p) < 0 for 0 ≤ t < T0, hence (44.44).

Step 2. Set J = ur + ηruq. We claim that for

η = C(n, p, q)β1−q , (44.45)

with C(n, p, q) > 0 sufficiently small, there holds

J ≤ 0 in Q := (0, 1/2)× (0, T1). (44.46)

We compute

Jt − Jrr =(ut − urr

)r+η(r(uq)t − (ruq)rr

)=(−n− 1

r2+ k(t)pup−1

)ur +

n− 1r

urr

+ η(qruq−1(ut − urr)− 2quq−1ur − q(q − 1)ruq−2(ur)2

)≤(−n− 1

r2+ k(t)pup−1 + (n− 3)qηuq−1

)ur

+n− 1

r

(J − ηruq

)r+qηk(t)rup+q−1

=(−n− 1

r2+ k(t)pup−1 − 2qηuq−1

)(J − ηruq

)+

n− 1r

Jr − ηn− 1

ruq + qηk(t)rup+q−1

= a(r, t)J +n− 1

rJr + b(r, t),

wherea(r, t) = −n− 1

r2+ k(t)pup−1 − 2qηuq−1

andb(r, t) = ηrup+q−1

((q − p)k(t) + 2qηuq−p

).

Using the definition of T1, (44.42) and (44.45), we obtain

b(r, t) ≤ ηrup+q−1(2qηβq−p − 2(p− q)Kβ1−p

)≤ 0 in Q.

On the other hand, for t ∈ [0, T1), we have J(0, t) = 0 and, by (44.43) and (44.44),the choice (44.45) implies J(1/2, t) ≤ 0. Also, for t = 0, using (44.37), u0 ≥ β andp > q, (44.45) implies J(r, 0) ≤ 0 in [0, 1/2]. Since a is bounded from above in(0, 1/2) × (0, τ) for each τ < T1, Claim (44.46) thus follows from the maximumprinciple (see Proposition 52.4).

By integrating (44.46), we have (u1−q)r ≥ (q − 1)ηr in (0, 1/2]× (0, T1). Thiscombined with (44.43) yields (44.41).

Proof of Theorems 44.5(ii) and 44.6. For M > 0, let β and u0 be as in Lem-ma 44.12. Since p > psg, we may fix q such that 1 + 2p/n < q < p. We deducefrom Lemma 44.12 that∫

|x|≤1

up(t) ≤ C(n, p)βp

∫ 1

0

rn−1−2p/(q−1) dr = C(n, p)βp, 0 < t < T1.

Taking 0 < β ≤ β0(n, p) sufficiently small, we infer that

k(t) ≥ C(n, p)β−p ≥ 4K(n, p)β1−p = 4λ, 0 < t < T1.

Consequently T1 = T0 = min(T, β). In particular, by the comparison principle(use Proposition 52.7), it follows that u ≥ w for t < min(T, β, Tw). But we haveTw < β for M large by Lemma 44.11, and we know that w blows up in L∞-norm.It follows that T ≤ Tw <∞, which proves Theorem 44.5(ii).

Since T1 = T , the first part of Theorem 44.6 is now a direct consequence ofLemma 44.12.

Finally, let us show that estimate (44.26) cannot be satisfied for any ε < 0.Suppose the contrary. This implies

supt∈(0,T )

‖u(t)‖q < ∞ for some q > n(p− 1)/2. (44.47)

On the other hand, u is bounded on ST and, by (44.25) and Holder’s inequality,we have

k(t) ≤ C, 0 ≤ t < T. (44.48)

Owing to (44.47), by comparison argument with (a variant of) the model problem(14.1), it follows from Theorem 16.4 (or, alternatively, Theorem 15.2 or Exam-ple 51.27 in Appendix E) that u is uniformly bounded in QT : a contradiction.

Proof of Theorem 44.7. (i) Due to (44.48), the lower estimate follows fromProposition 44.3(i).

(ii) In view of Proposition 44.3(ii), to prove the upper estimate, it suffices toshow that

k(t)→ ∈ (0,∞), as t→ T . (44.49)

Using (44.48), (44.26), parabolic estimates and the embedding (1.2), for someν ∈ (0, 1) we have u ∈ BUCν(γ < |x| < 1 − γ × (T/2, T )) for each γ > 0.Consequently, for all x ∈ B(0, 1) \ 0, limt→T u(x, t) exists and is finite. Since2p/(p−1) < n, using (44.26), the dominated convergence theorem and (44.48), wededuce (44.49).

Remark 44.13. By the methods of this subsection, problem (44.24) with f(u) =up and g(u) = uq can be studied for more general values of p, q > 1 and m ∈ R,under either Neumann or Dirichlet boundary conditions.

44.3. A problem with variational structure

We next consider equation (44.1) with

f(u) = |u|p−1u, g(u) = λ +∫ u

0

f(s) ds = λ +|u|p+1

p + 1,

where p > 1, m = −q < 0 and λ > 0, under Dirichlet boundary conditions. Takingλ = |Ω|−1 for simplicity, this leads to the problem

ut −∆u =(

1 +∫

Ω

|u(y, t)|p+1

p + 1dy

)−q

|u|p−1u, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (44.50)

where u0 ∈ L∞(Ω).

Problem (44.50) possesses a variational structure. Namely, the energy functional

E(u) =12

∫Ω

|∇u|2 dx − 11− q

(1 +

∫Ω

|u|p+1

p + 1dx

)1−q

(for q = 1, with an obvious modification if q = 1) is nonincreasing along anysolution of (44.50). More precisely

d

dtE(u(t)

)= −

∫Ω

u2t (t) dx

(this follows in the same way as in (17.7) and Example 51.28).

Theorem 44.14. Consider problem (44.50) with Ω bounded, p > 1 and 0 < q <(p − 1)/(p + 1). There exists C = C(p, q) > 0 such that, if u0 ∈ L∞ ∩ H1

0 (Ω)satisfies E(u0) < −C, then Tmax(u0) < ∞.

Proof. Set ψ(t) := ‖u(t)‖22. Multiplying the equation in (44.50) by u we obtain

12ψ′(t) =

∫Ω

uut(t) dx = −∫

Ω

|∇u(t)|2 dx +(

1 +∫

Ω

|u|p+1

p + 1dx

)−q ∫Ω

|u|p+1 dx

= −2E(u(t)

)+(

1 +∫

Ω

|u|p+1

p + 1dx

)−q

×(

(p− 1)− q(p + 1)(p + 1)(1− q)

∫Ω

|u|p+1 dx− 21− q

)≥ −2E(u0) + c1

(1 +

∫Ω

|u|p+1

p + 1dx

)1−q

− c2,

where c1, c2 > 0 depend only on p, q. Applying Holder’s inequality, we obtain

ψ′ ≥ cψγ − 2E(u0)− c2

with c = c(p, q, Ω) > 0 and γ := (p + 1)(1 − q)/2 > 1. If E(u0) < −c2/2 (orψγ(0) > 2(E(u0) + c2)/c), then this inequality implies Tmax(u0) <∞.

Remark 44.15. A priori bounds. Results on boundedness and a priori esti-mates of global solutions and universal bounds for global nonnegative solutions forproblems of the form (44.50) have been proved in [187], [440], [464].

44.4. A problem arising in the modeling of Ohmic heating

We finally consider equation (44.1) with f(u) = λe−u, g(u) = e−u, λ > 0, m = −2,n = 1 and Ω = (−1, 1), under Dirichlet boundary conditions. Namely:

ut − uxx = λ(∫ 1

−1

e−u(y, t) dy)−2

e−u, x ∈ (−1, 1), t > 0,

u(±1, t) = 0, t > 0,

u(x, 0) = u0(x), x ∈ (−1, 1),

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (44.51)

where we assume u0 ∈ L∞(Ω).Problem (44.51) arises from a special case of the following elliptic-parabolic

coupled system:

ut −∆u = σ(u)|∇φ|2, x ∈ Ω, t > 0,

div(σ(u)∇φ) = 0, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

φ = φ0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(44.52)

Here u and φ respectively represent the temperature and the electric potentialin a thermistor, i.e. a conductor whose electric conductivity σ = σ(u) may varywith the temperature, and the RHS of the first equation in (44.52) stands forthe heat production due to the Joule effect. We refer to [33] and the referencestherein for results on blow-up and global existence concerning system (44.52) andits variants. Now assume that the (thin) conductor can be represented by theinterval x ∈ (−1, 1) and that the potential φ is imposed to be 0 at x = −1 anda constant V at x = 1. The second equation in (44.52) becomes (σ(u)φx)x = 0and can be integrated in σ(u)φx = I(t) (which represents the electric currentper cross-sectional area unit). Denoting ρ(u) = 1/σ(u) (electric resistivity), henceφx = ρ(u)I(t), we obtain V =

∫ 1

−1 φx dx = I(t)∫ 1

−1 ρ(u) dx. The first equation in(44.52) then rewrites as

ut − uxx = σ(u)|φx|2 = ρ(u)I2(t) = ρ(u)V 2(∫ 1

−1

ρ(u) dx)−2

.

If the conductivity law is given by σ(u) = eu and thermal cooling is applied onthe ends of the conductor, we thus arrive at problem (44.51).

We shall see that the global behavior of solutions to (44.51) is closely relatedto the properties of the stationary problem

wxx + λ(∫ 1

−1

e−w dy)−2

e−w = 0, x ∈ (−1, 1), with w(±1) = 0. (44.53)

Proposition 44.16. Let λ > 0. Problem (44.53) has a (classical) solution if andonly if λ < 8. Moreover the solution is unique and it is given by wλ = zα, where

zα(x) = 2 log(cos(αx)

cosα

), (44.54)

and α ∈ (0, π/2) satisfies λ = 8 sin2 α. Furthermore, for |x| < 1, zα(x) is anincreasing function of α, hence wλ(x) is an increasing function of λ.

Proof. Setting µ = λ(∫ 1

−1e−w dy

)−2, we see that w solves

zxx + µe−z = 0, x ∈ (−1, 1), with z(±1) = 0. (44.55)

By direct calculation, we see that a solution of (44.55) is given by (44.54) where αis the unique number in (0, π/2) such that µ = 2α2/ cos2 α. On the other hand, thesolution of (44.55) is unique. Indeed, if y and z are two solutions, by subtractingthe equations for y and z and multiplying by z − y, we get

0 =∫ 1

−1

((yxx − zxx) + µ(e−y − e−z)

)(z − y) dx ≥

∫ 1

−1

(yx − zx)2 dx,

hence y = z. Finally, since∫ 1

−1 e−zα dy = 2α sin α cosα, the function zα solves

(44.53) with λ = 8 sin2 α, hence the necessary and sufficient condition on λ. Theremaining assertion follows from ∂

∂αzα(x) = 2(tanα− x tan(αx)).

The following result is from [314].

Theorem 44.17. Consider problem (44.51) with λ > 0 and u0 ∈ L∞(Ω).(i) If λ < 8, then the equilibrium wλ is globally asymptotically stable. Namely, allsolutions are global and converge to wλ in L∞(−1, 1) as t→∞.(ii) If λ > 8, then all solutions blow up in finite time. Moreover, the blow-up isglobal, i.e.

limt→Tmax(u0)

u(x, t) = ∞, −1 < x < 1. (44.56)

(iii) If λ = 8, then all solutions are global and unbounded. Moreover,

limt→∞u(x, t) =∞, −1 < x < 1.

Problem (44.51) admits a comparison principle (cf. Proposition 52.25). Theproof of Theorem 44.17 is based on suitable sub- and supersolutions, which willbe constructed under a “quasi-stationary” form (cf. (44.59) below).

Proof of Theorem 44.17. By a time shift we may assume without loss of gen-erality that

u0 ∈ C1([−1, 1]) and u0(±1) = 0. (44.57)

Also, denoting by ϕ1 the first eigenfunction, we observe that:

if Tmax(u0) =∞, then u(x, t) ≥ εϕ1(x) for some ε > 0 small and all t large.(44.58)

Indeed, fixing C > 0 such that u0 ≥ −Cϕ1, one easily checks that u(x, t) :=(2ε − (C + 2ε)e−λ1t

)ϕ1(x) is a subsolution to problem (44.51) for ε > 0 small

enough.In order to construct sub-/supersolutions, we put

v(x, t) = zα(t)(x), (44.59)

where α(t) is a function to be determined. Plugging (44.54), (44.59) into equation(44.51), we compute

Pv := vt − vxx − λ(∫ 1

−1

e−v dy)−2

e−v

= 2(tan α− x tan(αx)

)α′ +

2α2

cos2(αx)− λα2

4 sin2 α cos2(αx)

= 2(tan α− x tan(αx)

)α′ +

(8 sin2 α− λ)α2

4 sin2 α cos2(αx).

If λ < 8, then define α ∈ (0, π/2) by λ = 8 sin2 α (hence wλ = zα); otherwiseset α := π/2.

We first assume λ < 8 and look for a decreasing supersolution v = v. Thustaking α′ ≤ 0, we have

Pv ≥ 2α′ tan α +(8 sin2 α− λ)α2

4 sin2 α(44.60)

provided 8 sin2 α(t) ≥ λ. Due to (44.57), we may choose some α0 ∈ (α, π/2) closeenough to π/2 so that zα0(x) ≥ u0(x). Let α(t) be the solution of the ODE

α′(t) =(λ− 8 sin2 α)α2 cosα

8 sin3 α, t ≥ 0 (44.61)

with α(0) = α0. Since 8 sin2 α0 > λ, it is clear that α exists globally and satisfiesα′ < 0 and limt→∞ α(t) = α. It follows from (44.60) that v is a supersolution toproblem (44.51). Consequently

u ≤ v, 0 < t < Tmax(u0), (44.62)

hence in particular Tmax(u0) =∞. Moreover there holds

limt→∞ v(x, t) = wλ(x) uniformly in [−1, 1]. (44.63)

Now consider general λ again. Looking for an increasing subsolution v = v,hence α′(t) ≥ 0, we have

Pv ≤ 2α′ tan α +(8 sin2 α− λ)α2

4 sin2 α(44.64)

provided 8 sin2 α ≤ λ. Assuming Tmax(u0) =∞ and using (44.58), we may choosesome α1 ∈ (0, α) small enough so that zα1(x) ≤ u0(x). Take now α(t) to be thesolution of (44.61) with α(0) = α1. Since 8 sin2 α1 < λ, it is clear that α existsglobally and satisfies α′ > 0. Moreover, we have

limt→∞ α(t) = α.

It follows from (44.64) that v is a subsolution to problem (44.51). Consequently

u ≥ v, 0 < t < Tmax(u0).

If λ < 8, there holds in addition limt→∞ v(x, t) = wλ(x), uniformly in [−1, 1].This, along with (44.62), (44.63), proves assertion (i).

If λ ≥ 8, we have shown that either Tmax(u0) <∞, or

Tmax(u0) = ∞ and u(x, t) ≥ 2 log(cos(α(t)x)

cosα(t)

)→∞, t→∞. (44.65)

Assume λ > 8. We shall show by a further subsolution argument that (44.65)leads to a contradiction. We look for a modified subsolution of the form

v(x, t) = p log(cos(αx)

cosα

)where the function

α : [0, T0)→ [α2, π/2) (44.66)

and the numbers p > 1, T0 > 0, α2 ∈ (0, π/2) are to be determined.

We shall use the following elementary lemma:

Lemma 44.18. For each p > 1 and ε > 0, there holds

I(a) :=∫ 1

−1

dy

cosp(ay)≤ 4 + ε

π(p− 1) cosp−1 a, as a → (π

2 )−. (44.67)

Proof. We write

12I(a) =

∫ 1

0

dy

cosp(a(1− y))=∫ 1

0

dy

sinp(π2 − a + ay)

≤ 1sinp(aη)

+∫ η

0

dy


(44.68)

for 0 < a < π2 and 0 < η < 1. Fix η = η(ε) > 0 small. Taking 0 < π/2 − a < η,

and using sinx ∼ x as x→ 0, we obtain∫ η

0

dy


≤ (1 + ε/8)∫ η

0

dy

(π2 − a + ay)p

=1 + ε/8a(1− p)

[(π

2− a + ay

)1−p]η0≤ 1 + ε/8

a(p− 1)

(π

2− a)1−p

.

Since cos a ∼ (π2 − a) as a → π/2, this combined with (44.68) and

lima→π/2

cosp−1 a

sinp(aη)= 0

yields (44.67).

Proof of Theorem 44.17 (continued). Assuming α′(t) ≥ 0, we have

Pv ≤ pα′ tanα +pα2

cos2(αx)− λ

cosp α cosp(αx)I2(α).

For p ∈ (1, 2), by using (44.66), (44.67) and taking α2 close to π/2, we have

pα2

cos2(αx)− λ

cosp α cosp(αx)I2(α)≤ pα2

cos2(αx)− λπ2(p− 1)2 cosp−2 α

(4 + ε)2 cosp(αx)

≤ π2 cosp−2 α

cosp(αx)

(p

4− λ(p− 1)2

(4 + ε)2).

Since λ > 8, we can choose p ∈ (1, 2) close to 2 and ε small such that

γ := π2(λ(p− 1)2

(4 + ε)2− p

4

)> 0.

Taking α2 still closer to π/2 and using tan a ∼ (π2 −a)−1 as a → (π/2)−, it follows

that

Pv ≤ pα′ tanα− γcosp−2 α

cosp(αx)≤ 2p

(π

2− α

)−1

α′ − γ(π

2− α

)p−2

. (44.69)

Take now α(t) to be the solution of

α′(t) =γ

2p

(π

2− α

)p−1

, α(0) = α2.

Since 1 0. On the other hand, owing to (44.65), we may assumethat u0 ≥ v(·, 0) (after a time shift) which, along with(44.69), guarantees thatv is a subsolution to problem (44.51). Since limt→T0 v(x, t) = ∞ in (−1, 1), thiscontradicts Tmax =∞.

Let us finally prove global blow-up, i.e. (44.56). Denoting

h(t) =(∫ 1

−1

e−u dy)−2

and arguing as in the (alternative) proof of Proposition 23.1, we see that M(t) :=maxx∈[−1,1] u(x, t) satisfies

M ′(t) ≤ g(t) := λh(t)e−M(t), for a.e. 0 < t < T := Tmax(u0). (44.70)

Since u ≥ min[−1,1] u0 by the maximum principle, T < ∞ implies lim supt→T M(t)= ∞. Integrating (44.70), we deduce that

∫ T

0g(t) dt = ∞. Since ut − uxx =

λh(t)e−u ≥ g(t), (44.56) then follows from Theorem 43.6(i). This completes theproof of assertion (ii).

As for the critical case λ = 8, global existence can be shown by a modifiedsupersolution argument. We refer for this to [314].

Remarks 44.19. (a) Formal results concerning the blow-up rate (and the behav-ior in the boundary layer) for problem (44.51) are given in [315].

(b) Results on problem (44.51) with more general conductivity functions σ(u)can be found in [315], [66]. For the analogue of problem (44.51) in dimension n = 2,results similar to Theorem 44.17 are proved in [518], [298] for radial solutions in adisk.

(c) For problem (44.51) with Neumann boundary conditions, it is easy to seethat all solutions blow up in finite time: Indeed the solution of the ODE y′ = λ

4 ey

with y(0) = inf u0 is a subsolution.

45. Fujita-type results for problems involvingspace integrals

We consider Cauchy problems with nonlocal source terms involving space integrals,of the form

ut −∆u =(∫

Rn

K(y)uq(y, t) dy)(p−1)/q

u1+r, x ∈ Rn, t > 0,

u(x, 0) = u0(x), x ∈ Rn.

⎫⎬⎭ (45.1)

In what follows, we assume that

p > 1, q ≥ 1, r ≥ 0, u0 ∈ L∞(Rn), u0 ≥ 0,

K is a positive, bounded continuous function.(45.2)

If K ∈ L1(Rn), then we assume in addition, that u0 ∈ L1(Rn). Under theseassumptions, problem (45.1) is locally well-posed (see Example 51.13).

The critical exponent for problem (45.1) was studied in [227]. It will depend ina crucial way on whether or not the function K is integrable. In the integrablecase we have the following result.

Theorem 45.1. Assume (45.2), K ∈ L1(Rn), and let pc = 1 + 2n − r.

(i) If p < pc, then (45.1) admits no nontrivial global solution.(ii) If p > pc, then (45.1) admits both global positive and blowing-up solutions.

In the non-integrable case we need some additional assumptions on the asymp-totic behavior of K.

Theorem 45.2. Assume (45.2) and u0 ∈ L1(Rn). Assume in addition that Ksatisfies

c1(1 + |x|)−β ≤ K(x) ≤ c2(1 + |x|)−β , x ∈ Rn, (45.3)

45. Fujita-type results for problems involving space integrals 419

for some β ∈ [0, n) and some c1, c2 > 0. Let pc = 1 + q(2−nr)n(q−1)+β .

(i) If p < pc, then (45.1) admits no nontrivial global solution.(ii) If p > pc, then (45.1) admits both global positive and blowing-up solutions.

The proofs are exclusively based on comparison with suitable (self-similar) sub-and supersolutions. Note that (45.1) does admit a comparison principle (this fol-lows from Proposition 52.27). For results in the critical case p = pc, see [227].

Proof of Theorems 45.1 and 45.2. 1. Blow-up. We look for a blowing-upsubsolution under the form

u(x, t) = A(T − t)−αf(ξ), ξ =x√

T − t, f(ξ) = e−|ξ|2 ,

where α, T, A > 0 are parameters. We compute

ut = Aα(T−t)−α−1f(ξ)+A

2(T−t)−α−1ξ ·∇ξf(ξ), ∆u = A(T−t)−α−1∆ξf(ξ).

Denoting

I(t) =∫

Rn

K(y)e−q|y|2/(T−t) dy,

the condition for u being a subsolution is thus given by

αf +ξ

2· ∇ξf −∆ξf ≤ Ap+r−1(T − t)1−(r+p−1)αI

p−1q (t)f r+1,

that is

α + 2n ≤ 5|ξ|2 + Ap+r−1(T − t)1−(r+p−1)αIp−1

q (t)e−r|ξ|2 , ξ ∈ Rn, 0 < t < T.

(45.4)In the case K ∈ L1, assume without loss of generality that K ≥ c0χ|y|<ρ for

some c0, ρ > 0. We then have

I(t) = (T − t)n/2

∫Rn

K(z√

T − t)e−q|z|2 dz ≥ c0(T − t)n/2

∫|z|<ρ/

√T

e−q|z|2 dz,

henceI(t) ≥ C(T − t)n/2T−n/2

for all T ≥ 1 and some C > 0 (independent of T ).In the case when K satisfies (45.3), we have

I(t) ≥ C(T − t)n/2

∫Rn

(1 + |z|√

T − t)−βe−q|z|2 dz

≥ C(T − t)n/2T−β/2

∫ ∞

0

(T−1/2 + ρ)−βe−qρ2ρn−1 dρ,

henceI(t) ≥ C(T − t)n/2T−β/2

for all T ≥ 1 and some C > 0 (independent of T ).Let us now take

α =2q + n(p− 1)2q(p + r − 1)

and A = BT γ

with

γ =

⎧⎨⎩n(p−1)

2q(p+r−1) if K ∈ L1,

β(p−1)2q(p+r−1) if K satisfies (45.3).

A sufficient condition for (45.4) is then that

α + 2n ≤ 5|ξ|2 + c2Bp+r−1e−r|ξ|2 , ξ ∈ R

n.

This is satisfied for some large B > 0 and guarantees that u is a subsolution forall T ≥ 1.

Finally assume for contradiction that u exists for all time. Since u is a positivesupersolution of the linear heat equation, it follows that u(x, 1) ≥ εσ−n/2e−|x|2/4σ

for some ε, σ > 0, hence u(x, t + 1) ≥ ε(σ + t)−n/2e−|x|2/4(σ+t) for all t > 0(cf. (18.12)). Now, the assumption p < pc means that α− γ > n/2 in both cases.Taking T = 4(σ + t) and t > 0 sufficiently large, we thus get

u(x, t + 1) ≥ εT−n/2e−|x|2/T ≥ u(x, 0) = BT γ−αe−|x|2/T

and the comparison principle would then imply finite-time blow-up of u. Statement(i) of Theorems 45.1 and 45.2 follows.

2. Global existence. We look for a blowing-up supersolution under the form

u(x, t) = (T + t)−αg(ξ), ξ =x√

T + t, g(ξ) = e−σ|ξ|2 ,

where α, T, σ > 0 are parameters. We compute

ut = −α(T + t)−α−1g(ξ)− 12(T + t)−α−1ξ · ∇ξg(ξ), ∆u = (T + t)−α−1∆ξg(ξ).

Denoting

J(t) =∫

Rn

K(y)e−qσ|y|2/(T+t) dy,

the condition for u being a supersolution is thus given by

−αg − ξ

2· ∇ξg −∆ξg ≥ (T + t)1−(r+p−1)αJ

p−1q (t)gr+1.

46. A problem with memory term 421

Taking σ = 1/4, this amounts to

n

2− α ≥ (T + t)1−(r+p−1)αJ

p−1q (t)e−rσ|ξ|2 , ξ ∈ R

n, t > 0. (45.5)

In the case K ∈ L1, there obviously holds J(t) ≤ ‖K‖L1. Taking

1p + r − 1

< α <n

2

and T large, we obtain (45.5).In the case when K satisfies (45.3), since β < n, we have

J(t) ≤ C(T + t)n/2

∫Rn

(1 + |z|√

T + t)−βe−qσ|z|2 dz

≤ C(T + t)(n−β)/2

∫ ∞

0

|z|−βe−qσ|z|2 dz = C(T + t)(n−β)/2.

Since p > pc, we may take

2q + (n− β)(p− 1)2q(p + r − 1)

< α <n

2

and T large yields

n

2− α ≥ C(T + t)1−α(p+r−1)+(n−β)(p−1)/2q, t ≥ 0,

hence (45.5)In either case, we have thus shown that u is a supersolution, which implies the

global existence of u whenever 0 ≤ u(0) < u(0).

46. A problem with memory term

We consider the following problem

ut −∆u =∫ t

0

up(x, s) ds − kuq, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (46.1)

where Ω is bounded, p > 1, q ≥ 1, k ≥ 0, and u0 ∈ L∞(Ω), u0 ≥ 0. Notice thatthe problem is well-posed due to Example 51.14.

The following result [479] shows that q = p constitutes a critical blow-up exponentfor problem (46.1). Moreover, blow-up (in finite or infinite time) occurs for allpositive solutions of (46.1), and not only for solutions with large initial data,unlike in problems (43.1) and (15.1) for instance.

Theorem 46.1. Consider problem (46.1) with Ω bounded, p > 1, q ≥ 1, k ≥ 0,and 0 ≤ u0 ∈ L∞(Ω), u0 ≡ 0.(i) If p > q or k = 0, then all solutions of (46.1) blow up in finite time.(ii) If p ≤ q and k > 0, then all solutions of (46.1) are global and unbounded, thatis, lim supt→∞ ‖u(t)‖∞ = ∞.

The proof of Theorem 46.1 relies on the eigenfunction method (cf. the proofof Theorem 17.1), combined with the following lemma concerning the system ofdifferential inequalities

z′ ≥ yp,

y′ + λy + kz′r ≥ z.

(46.2)

Lemma 46.2. Assume 0 < r < 1 0 and (46.2) on (0, T ). Then T < ∞.

Proof. By translating the origin of time, we may assume that actually y, z ∈C1([0, T )) and z(0) > 0. Fix γ such that max(r, 1/p) < γ < 1. It follows from thefirst inequality in (46.2) that, for all ε > 0, there exists a constant Cε > 0 suchthat

Cεz′γ ≥ ypγ + (3λ + 1)y − ε and Cεz

′γ ≥ 3kz′r − ε,

hence2Cεz

′γ + 3y′ ≥ 3(y′ + λy + kz′r

)+ypγ + y − 2ε.

By the second inequality in (46.2), we deduce that

2Cεz′γ + 3y′ ≥ 3z + ypγ + y − 2ε. (46.3)

Next take m ∈ (0, γ). By Young’s inequality, we have

2Cεz′γ = 2Cε

z′γ

zmzm ≤ εzm/(1−γ) + C′

ε

z′

zm/γ,

hence

C′′ε (zθ)′ + εzm/(1−γ) ≥ 2Cεz

′γ , where θ = 1− (m/γ) ∈ (0, 1), (46.4)

for some large constant C′′ε > 0.

Now assume further that m < 1− γ and define

φ = C′′ε zθ + 3y.

By combining (46.3) and (46.4), for ε < 1, we get

φ′ ≥ 3z + ypγ + y − 2ε− εzm/(1−γ) ≥ 2z + ypγ + y − 3ε, 0 ≤ t < T.

Choosing ε < z(0)/3, setting ν = min(pγ, 1/θ) > 1 and using the fact that z isnondecreasing, we then obtain

φ′ ≥ z + ypγ + y ≥[z(0)

]1−θνzθν + yν ≥ Cφν

on (0, T ), for some C > 0. We conclude that T < ∞.

Proof of Theorem 46.1. (i) Define the functions

y(t) =∫

Ω

u(x, t)ϕ1(x) dx and z(t) =∫ t

0

∫Ω

up(x, s)ϕ1(x) dx ds, 0 ≤ t < T.

Multiplying (46.1) by ϕ1 and integrating by parts over Ω, we get:

y′ + λ1y =∫ t

0

∫Ω

up(x, s)ϕ1(x) dx ds − k

∫Ω

uq(x, t)ϕ1(x) dx, 0 < t < T.

We may assume q < p also if k = 0. Letting r = q/p < 1 and applying Jensen’sinequality yields

y′ + λ1y + kz′r ≥ z and z′ ≥ yp.

The conclusion thus follows from Lemma 46.2.(ii) If p < q, a simple calculation shows that v(x, t) = C(1 + t)1/(q−p) is a

supersolution for all large C > 0. If p = q, the same holds with v(x, t) = CeCt.Taking C > ‖u0‖∞, it follows from the comparison principle (Proposition 52.25)that u must exist globally.

Last, assume for contradiction that u is globally bounded by a constant M > 0.Then u satisfies

ut −∆u ≥∫ t

0

up(x, s) ds− aM q−1u, x ∈ Ω, t > 0.

By the comparison principle, in view of part (i), this immediately implies finite-time blow-up: a contradiction.

Remarks 46.3. (i) The assumption r < 1 in Lemma 46.2 is essential, at least ifk > 0. Indeed, if r = 1, then z(t) = Ceµt, y(t) = (Cµ)1/peµt/p is a global positivesolution of (46.2) for µ = 1/k and any C > 0.

(ii) Fujita-type results. For problem (46.1) and related equations, Fujita-typeresults have been recently obtained in [112].

46.2. Blow-up rate

The following result shows a type I blow-up rate for monotone-in-time solutionsand provides a sufficient condition for monotonicity.

Theorem 46.4. Consider problem (46.1) with Ω bounded, p > 1, k = 0. Letu0 ∈ C1(Ω), u0 ≥ 0, u0 ≡ 0, and T := Tmax(u0).(i) Assume that:

there exists t0 ∈ [0, T ) such that ut(x, t0) ≥ 0 for all x ∈ Ω. (46.5)

Then T <∞, ut ≥ 0 in Ω× [t0, T ) and u satisfies the blow-up estimate

C1(T − t)−2/(p−1) ≤ ‖u(t)‖∞ ≤ C2(T − t)−2/(p−1), as t→ T . (46.6)

(ii) Assume that Φ ∈ C2(Ω) satisfies Φ > 0 in Ω, Φ|∂Ω = 0 and that there existε, η > 0 such that

∆Φ(x) ≥ εδ(x) for all x ∈ Ω such that δ(x) ≤ η. (46.7)

Then, for all λ > 0 large enough, the solution of (46.1) with initial data u0 = λΦsatisfies (46.5).

Part (i) was proved in [335] (under the additional assumption Ω = BR andu0 radially symmetric decreasing). Part (ii) was proved in [486]. Note that (46.5)cannot be satisfied for 0 ≤ t0 T , due to ut(., 0) = ∆u0,

Proof. (i) Let

J(x, t) = ut − ε

∫ t

0

up ds, (x, t) ∈ Ω× (t1, T ).

By Example 51.14 we have J ∈ C2,1(QT )∩C(Ω× (0, T )). Pick t1 ∈ (t0, T ). Takingε > 0 small enough, using (46.5) and arguing as in the proof of Theorem 23.5, thistime using the nonlocal maximum principle in Proposition 52.24, we obtain thatJ(·, t1) ≥ 0 in Ω. We compute

Jt −∆J = utt − εup −∆ut + εp

∫ t

0

up−1∆u ds + εp(p− 1)∫ t

0

up−2|∇u|2 ds

≥ (1 − ε)up + εp

∫ t

0

up−1(ut −

∫ s

0

up dσ)

ds

= (1 − ε)up0 + p

∫ t

0

up−1(ut − ε

∫ s

0

up dσ)

ds ≥ p

∫ t

0

up−1J ds.

Since J = 0 on ∂Ω × (t1, T ), it follows from Proposition 52.24 that J ≥ 0 inΩ× (t1, T ).

Now, for each fixed x ∈ Ω, multiplying the inequality ut ≥ ε∫ t

t1up ds by up and

integrating over (t1, t), we obtain

up(x, t) ≥ c(∫ t

t1

up ds)2p/(p+1)

, t1 < t < T.

It follows that T <∞ and, by integrating over (t, T ), we obtain∫ t

t1

up ds ≤ C(T − t)−(p+1)/(p−1), t1 < t < T. (46.8)

Setting t′ = t + (T − t)/2 and using ut ≥ 0, we deduce that

T − t

2up(x, t) ≤

∫ t′

t

up ds ≤ C(T − t′)−(p+1)/(p−1) = C(T − t

2

)−(p+1)/(p−1)

,

henceu(x, t) ≤ C(T − t)−2/(p−1), t1 < t < T,

with C = C(p, ε). The upper estimate in (46.6) follows.On the other hand, letting M(t) = ‖u(t)‖∞ and arguing as in the (alternative)

proof of Proposition 23.1, we get

M ′(t) ≤∫ t

0

Mp(s) ds, for a.e. 0 < t < T .

Proceeding similarly as for (46.8), we obtain∫ t

0

Mp ds ≥ c1(T − t)−(p+1)/(p−1), 0 < t < T. (46.9)

For t1 ≤ τ < t < T , by using (46.9), the upper estimate in (46.6) and M beingnondecreasing on [t1, T ), we obtain

c1(T − t)−(p+1)/(p−1) ≤∫ τ

0

Mp ds +∫ t

τ

Mp ds

≤ C(T − τ)−(p+1)/(p−1) + (t− τ)Mp(t).

For t close enough to T , taking τ = T − γ(T − t) with γ = (2C/c1)(p−1)/(p+1) > 1,we get,

M(t) ≥ (c1/2γ)1/p(T − t)−2/(p−1),

which proves the lower estimate.(ii) Let v = vλ := ut. By Example 51.14 we have v ∈ C2,1(QT )∩C([0, T ), L2(Ω)).

The function v satisfies

vt −∆v = up, x ∈ Ω, 0 < t < T,

v = 0, x ∈ ∂Ω, 0 < t < T,

v(x, 0) = ∆u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭hence

v(t) = e−tA(∆u0) +∫ t

0

e−(t−s)Aup(s) ds.

Since u(t) ≥ e−tAu0 and


)p ≥ (e−(t−s)A(e−sAu0))p =

(e−tAu0

)p,

we have

v(t) ≥ e−tA(∆u0) +∫ t

0


)pds ≥ e−tA(∆u0) + t

(e−tAu0

)p.

Therefore, for all λ > 0,

1λ

vλ(t) ≥ e−tA(∆Φ) + λp−1t(e−tAΦ

)p, 0 ≤ t < T (λΦ). (46.10)

We claim that there exists η1 > 0 such that

e−tA(∆Φ)(x) > 0 for all (x, t) such that δ(x) ≤ η1 and 0 ≤ t ≤ η1. (46.11)

To prove the claim, observe that, by the assumption (46.7), there exist γ > 0and ρ ∈ D(Ω), ρ ≥ 0, such that ∆Φ ≥ γϕ1 − ρ in Ω. Therefore, e−tA(∆Φ) ≥γe−λ1tϕ1 − e−tAρ. Using ϕ1 ≥ c1δ(x) in Ω, ρ ∈ D(Ω) and the continuity of e−tAρin C1(Ω) at t = 0, claim (46.11) follows.

A straightforward calculation shows that w(t) :=(‖u0‖−(p−1)/2

∞ −kt)−2/(p−1) is

a supersolution of (46.1) for k = (p− 1)(2(p + 1))−1/2. Since blow-up takes placein L∞-norm if it occurs, this implies in particular that T (u0) ≥ 1

k‖u0‖−(p−1)/2∞ .

Taking t = tλ := 12k‖λΦ‖−(p−1)/2

∞ in (46.10), we obtain

1λ

vλ(tλ) ≥ e−tλA(∆Φ) +12k‖Φ‖−(p−1)/2

∞ λ(p−1)/2(e−tλAΦ

)p.

On the one hand, since Φ ≥ 0, by (46.11) we have

1λ

vλ(x, tλ) ≥ 0 if δ(x) ≤ η1 and λ ≥ λ0(p, Φ) > 0 large enough.


On the other hand, since Φ > 0 in Ω, we have e−tAΦ > 0 in Ω × [0,∞) by thestrong maximum principle. Therefore, there exists α > 0 such that

(e−tAΦ

)(x) ≥ α

for all (x, t) such that δ(x) ≥ η1 and t ∈ [0, 1]. It follows that if δ(x) ≥ η1 andλ ≥ λ0(p, Φ) (possibly larger), then

1λ

vλ(x, tλ) ≥ −‖∆Φ‖∞ + C(p, Φ)αpλ(p−1)/2 > 0.

We have thus shown that ut(x, tλ) ≥ 0 in Ω whenever λ ≥ λ0(p, Φ) and the theoremis proved.

Remark 46.5. Blow-up set and profiles. Results on single-point blow-up andon blow-up profiles for equations similar to (46.1) have been obtained in [67] byemploying methods from [219].

Appendices

47. Appendix A: Linear elliptic equations

In this appendix we collect some fundamental estimates for linear elliptic equa-tions.

47.1. Elliptic regularity

We assume that Ω is an arbitrary domain in Rn and we consider second-order

elliptic differential operators of the form

Au = −n∑

i,j=1

aij∂2

∂xi∂xju +

n∑i=1

bi∂

∂xiu + cu, (47.1)

with measurable coefficients aij , bi, c satisfying the ellipticity condition∑i,j

aij(x)ξiξj ≥ λ|ξ|2 for all x ∈ Ω, ξ ∈ Rn, (47.2)

with λ > 0 and a uniform bound

|aij |, |bi|, |c| ≤ Λ. (47.3)

We consider the linear problem

Au = f in Ω, (47.4)

where f = f(x) is a given function.

A strong solution of (47.4) is a function u ∈ W 2,1loc (Ω) which satisfies (47.4)

a.e. We denote by ‖u‖k,p;D the norm in W k,p(D); in particular ‖u‖k,p;Ω = ‖u‖k,p.The following result (cf. [250, Theorems 9.11 and 9.13]) contains the basic in-

terior and interior-boundary elliptic Lp-estimates.

Theorem 47.1. Let Ω be an arbitrary domain in Rn and assume (47.2) and

(47.3). Let u ∈W 2,ploc ∩Lp(Ω), 1 < p < ∞, be a strong solution of (47.4), where aij

are continuous and f ∈ Lp(Ω).(i) Consider a subdomain Ω′ ⊂⊂ Ω. Then

‖u‖2,p;Ω′ ≤ C(‖u‖p + ‖f‖p), (47.5)

430 Appendices

where C depends only on n, p, Ω, Ω′, λ, Λ, and the moduli of continuity of the aij

on Ω′.

(ii) Let Σ be an open subset of ∂Ω of class C2, u ∈ W 2,p(Ω) and u = 0 on Σ inthe sense of traces. Let aij ∈ C(Ω ∪ Σ) and Ω′ ⊂⊂ Ω ∪ Σ. Then (47.5) is true,where C depends also on Σ.

As for interior and interior-boundary elliptic Schauder estimates, we have thefollowing theorem (cf. [250, Corollary 6.3, Theorems 6.6, 6.19 and Lemma 6.16]).

Theorem 47.2. Let Ω be an arbitrary bounded domain in Rn, assume (47.2), and

let f and the coefficients of A belong to BUCα(Ω), where α ∈ (0, 1).

(i) Consider a subdomain Ω′ ⊂⊂ Ω. If u ∈ C2(Ω) is a solution of (47.4), thenu ∈ BUC2+α(Ω′) and

‖u‖BUC2+α(Ω′) ≤ C(‖u‖∞ + ‖f‖BUCα(Ω)

),

where C depends only on n, α, λ, Ω, Ω′ and the norms of the coefficients of A inBUCα(Ω).

(ii) Assume Ω of class C2+α and let ϕ ∈ BUC2+α(Ω). If u ∈ C2(Ω) ∩ C(Ω) is asolution of (47.4) satisfying u = ϕ on ∂Ω, then u ∈ BUC2+α(Ω) and

‖u‖BUC2+α(Ω) ≤ C(‖u‖∞ + ‖f‖BUCα(Ω) + ‖ϕ‖BUC2+α(Ω)

),

where C depends only on n, α, λ, Ω and the norms of the coefficients of A inBUCα(Ω).

If we deal only with weaker type of solutions (say, variational), then the regu-larity assumptions u ∈W 2,p(Ω) or u ∈ C2(Ω) in the above theorems can often beverified by means of the following existence-uniqueness theorem (cf. [250, Theo-rems 9.15 and 6.13]). See Remark 47.4(ii) for an example.

Theorem 47.3. Assume (47.2), (47.3), c ≤ 0 and let Ω be a bounded domain ofclass C2.

(i) Let aij ∈ C(Ω), f ∈ Lp(Ω) and ϕ ∈W 2,p(Ω), where 1 < p < ∞. Then equation(47.4) has a unique (strong) solution u ∈W 2,p(Ω) satisfying u− ϕ ∈ W 1,p

0 (Ω).

(ii) Let f and the coefficients of A belong to B(Ω) ∩ Cα(Ω), α ∈ (0, 1), and ϕ ∈C(∂Ω). Then equation (47.4) has a unique (classical) solution u ∈ C2+α(Ω)∩C(Ω)satisfying u = ϕ on ∂Ω.

Remarks 47.4. (i) Assume that A has constant coefficients. Then the followingregularity result can be deduced from Theorem 47.1(i): if u, f ∈ Lp

loc(Ω) for some1 < p < ∞ and Au = f in D′(Ω), then u ∈ W 2,p

loc (Ω). To show this, it suffices to

47. Appendix A: Linear elliptic equations 431

apply Theorem 47.1(i) to the convolution products u ∗ ρj , where ρj is a sequenceof mollifiers, i.e.

ρj(x) = jnρ(jx), 0 ≤ ρ ∈ D(Rn),∫

Rn

ρ(x) dx = 1 (47.6)

(see the end of the proof of Proposition 47.6 below for a more detailed, similar ar-gument). Similarly, using Theorem 47.2(i), we obtain that u ∈ C2+α(Ω) wheneveru, f ∈ Cα(Ω) for some 0 < α < 1 and Au = f is satisfied in D′(Ω).

(ii) For A with leading coefficients of class C1, Theorem 47.1(i) remains trueif we only assume that u ∈ W 1,p

loc ∩ Lp(Ω), equation (47.4) being understood inthe variational sense. The idea of the proof is as follows. Taking ψ a smooth cut-off function, the regularity of u and aij allows to apply Theorem 47.3(i) to theequation satisfied by the function uψ in a smooth domain Ω′ ⊂⊂ Ω. We can thenconclude by using the uniqueness of variational solutions.

(iii) As a useful consequence of the above theorems, we can prove the followingproperty. Assume that A has constant coefficients, let Ω ⊂ R

n be a (possiblyunbounded) domain of class C2, Σ an open subset of ∂Ω, and f ∈ Lp

loc(Ω ∪ Σ),with p > n. Assume that u ∈ W 2,p

loc (Ω) ∩ C(Ω ∪ Σ) satisfies Au = f a.e. in Ω andu = 0 on Σ. Then u ∈ W 2,p

loc (Ω ∪ Σ). If we further assume that Ω is of class C2+α

and that f ∈ Cα(Ω ∪ Σ) for some α ∈ (0, 1), then u ∈ C2+α(Ω ∪ Σ).Let us prove this in the case A = −∆ for simplicity. Let x0 ∈ Σ. One can find

r > 0 and a bounded domain ω, as smooth as Ω, such that Ω ∩B(x0, r) ⊂ ω ⊂ Ωand ∂Ω∩B(x0, r) ⊂ Σ. Let ϕ ∈ D(Rn) be such that supp(ϕ) ⊂ B(x0, r) and ϕ = 1near x = x0. Then v := uϕ satisfies

−∆v = f := fϕ− 2∇u · ∇ϕ− u∆ϕ in D′(ω). (47.7)

Since f ∈ W−1,p(ω), there exists a unique w ∈ W 1,p0 (ω) ⊂ C0(ω), such that

−∆w = f . Also, we have w ∈ W 2,ploc (ω) due to f ∈ Lp

loc(ω) and part (i). By themaximum principle in Proposition 52.1(i), we deduce that w = v. It follows thatu ∈ W 1,p

loc (Ω ∪ Σ). Getting back to equation (47.7), we now have f ∈ Lp(ω). ByTheorem 47.3(i) and the uniqueness of w, we deduce that w ∈ W 2,p(ω), henceu ∈W 2,p

loc (Ω ∪ Σ). Now, if also Ω ∈ C2+α and f ∈ Cα(Ω ∪ Σ), then f ∈ BUCβ(ω)with β = min(α, 1−n/p). By Theorem 47.2(ii), we get v ∈ BUC2+β(ω), hence u ∈C2+β(Ω∪Σ). Iterating, we finally obtain f ∈ BUCα(ω) and u ∈ C2+α(Ω∪Σ).

47.2. Lp-Lq-estimates

The following regularity results for the Laplacian are often used in bootstrap ar-guments in nonlinear problems. The notion of L1-solution of the Laplace equation

−∆u = f in Ω,

u = 0 on ∂Ω

(47.8)

has been introduced in Definition 3.1.

432 Appendices

Proposition 47.5. Let Ω ⊂ Rn be a bounded domain of class C2+α for some

α ∈ (0, 1), and assume that 1 ≤ p ≤ q ≤ ∞ satisfy

1p− 1

q<

2n

. (47.9)

Let f ∈ L1(Ω) and u be the L1-solution of (47.8).(i) If f ∈ Lp(Ω), then u ∈ Lq(Ω) and

‖u‖q ≤ C(Ω, p, q)‖f‖p.

(ii) If f+ ∈ Lp(Ω), then u+ ∈ Lq(Ω) and

‖u+‖q ≤ C(Ω, p, q)‖f+‖p.

Proposition 47.6. Let Ω be an arbitrary bounded domain in Rn, Ω′ ⊂⊂ Ω,

and assume that 1 ≤ p ≤ q ≤ ∞ satisfy (47.9). Let u ∈ L1(Ω) be such that−∆u =: f ∈ L1(Ω) (where ∆u is understood in the sense of distributions).(i) If f ∈ Lp(Ω), then u ∈ Lq(Ω′) and

‖u‖Lq(Ω′) ≤ C(Ω, Ω′, p, q)(‖f‖Lp(Ω) + ‖u‖L1(Ω)

). (47.10)

(ii) If f+ ∈ Lp(Ω), then u+ ∈ Lq(Ω′) and

‖u+‖Lq(Ω′) ≤ C(Ω, Ω′, p, q)(‖f+‖Lp(Ω) + ‖u+‖L1(Ω)

). (47.11)

Proposition 47.5 will be proved in Appendix C, along with the analogous resultin Lp

δ-spaces (Theorem 49.2). As for Proposition 47.6, for p > 1, inequality (47.10)with ‖u‖L1(Ω) replaced by ‖u‖Lp(Ω) would follow from Theorem 47.3(i) and theSobolev inequality. However, since we need the case p = 1 (and also (47.11))in the applications, we have to rely on different, classical arguments, using thefundamental solution and Green’s formula.

Proof of Proposition 47.6. We give the proof for n ≥ 3 only. The cases n = 1, 2can be treated similarly.

(i) We first assume that u is smooth, say C2. Fix x ∈ Ω′ and 0 < r < R :=min

(1, dist(Ω′, ∂Ω)

). Let Γr(y) = cn(|y|2−n−r2−n), where cn = ((n−2)|Sn−1|)−1,

be the fundamental solution of the Laplacian vanishing for |y| = r. It is well knownthat

u(x) =∫|y|<r

Γr(y)f(x + y) dy + (n− 2)cnr1−n

∫|y|=r

u(x + y) dσ, (47.12)

where σ denotes the surface measure of the sphere y ∈ Rn : |y| = r. (The

representation formula (47.12) follows by integrating by parts the function∆u(x + y)Γr(y) on the annulus ε < |y| < r and letting ε → 0, see e.g. [250,Section 2.4].) By integrating (47.12) in r over (R/2, R), we get

|u(x)| ≤ cnR

∫|y|<R

|y|2−n|f(x + y)| dy + C(n)R1−n

∫R/2<|y−x|<R

|u| dy

≤ cnR(h ∗ |f |

)(x) + C(n)R1−n‖u‖L1(Ω),

where h(y) := |y|2−nχ|y|<R and f denotes the extension of f by 0. Since h ∈Lr(Rn) for any r < n/(n− 2), the Young inequality for convolutions yields

‖h ∗ |f |‖q ≤ ‖h‖r‖f‖p = C‖f‖Lp(Ω), with 1− 1r

=1p− 1

q<

2n

.

Inequality (47.10) for smooth u follows.In the general case, let uj = u ∗ ρj , where ρj is a sequence of mollifiers defined

by (47.6), and fj := −∆uj = f ∗ ρj . Note that, for a given subdomain ω ⊂⊂ Ω,uj ∈ C∞(ω) for all j large enough. Choosing Ω′ ⊂⊂ Ω′′ ⊂⊂ Ω, we have

‖uj‖Lq(Ω′) ≤ C(‖fj‖Lp(Ω′′) + ‖uj‖L1(Ω′′)

)(47.13)

by the previous step. Using the facts that uj → u in L1(Ω′′) and fj → f in Lp(Ω′′),we may pass to the limit in (47.13) with help of Fatou’s lemma, and the conclusionfollows.

(ii) By (47.12), we have

u+(x) ≤∫|y|<r

Γr(y)f+(x + y) dy + (n− 2)cnr1−n

∫|y|=r

u+(x + y) dσ,

The rest of the proof is then similar.

We conclude this subsection by the following technical lemma, which makesprecise some relations between distributional and L1-solutions.

Lemma 47.7. Let Ω be an arbitrary bounded domain in Rn, ω ⊂⊂ Ω, and let

ψ ∈ D(Ω) be such that ψ = 1 in ω. If f ∈ L1loc(Ω) and u ∈ L1

loc(Ω) ∩ C1(Ω \ ω) isa solution of

−∆u = f in D′(Ω), (47.14)

then w := uψ is an L1-solution of

−∆w = f := fψ − 2∇u · ∇ψ − u∆ψ in Ω,

w = 0 on ∂Ω.

434 Appendices

Proof. Fix an open set U such that supp(ψ) ⊂ U ⊂⊂ Ω and let ϕ ∈ C2(Ω) ∩C∞(U). We write

−∫

Ω

uψ∆ϕdx = −∫

Ω

u∆(ψϕ) dx + 2∫

Ω

u∇ψ · ∇ϕ dx +∫

Ω

uϕ∆ψ dx.

Since ψϕ ∈ D(Ω) and u ∈ C1(Ω \ω), by using (47.14) and integrating by parts weobtain

−∫

Ω

w∆ϕ dx =∫

Ω

fψϕdx−∫

Ω

(2∇u · ∇ψ + u∆ψ

)ϕdx =

∫Ω

fϕ dx. (47.15)

Finally, since C2(Ω) ∩ C∞(U) is dense in C2(Ω), (47.15) remains true for all ϕ ∈C2(Ω) and the conclusion follows.

Remark 47.8. Proposition 47.5 remains true in case of equality in (47.9), pro-vided p > 1 and q < ∞. Indeed, noting that n ≥ 3, this follows from estimate(48.8) below and the Marcinkiewicz interpolation theorem.

47.3. An elliptic operator in a weighted Lebesgue space

In this part we prove some basic properties of weighted spaces L2g, H

1g , H2

g and theelliptic operator L defined below.

Let g(y) := e|y|2/4, y ∈ R

n,

Lqg := f ∈ Lq(Rn) :

∫Rn

|f(y)|qg(y) dy <∞,

H1g := f ∈ L2

g : ∇f ∈ L2g, H2

g := f ∈ H1g : ∇f ∈ H1

g and

Lv := −∆v − y · ∇v

2= −1

g∇ · (g∇v), v ∈ H2

g . (47.16)

Estimate (47.18) below (with u = ∂v/∂yi) shows that L : H2g → L2

g is a continuouslinear operator. We will consider L as an unbounded operator in the Hilbert spaceL2

g with domain of definition H2g . Notice that

(Lv, w)g =∫

Rn

(∇v · ∇w)g dy, v, w ∈ H2g ,

where (u, v)g :=∫

Rn uvg dy is the scalar product in L2g. Hence L is symmetric and

positive. The following two lemmas show that L is a self-adjoint operator withcompact inverse.


Lemma 47.9. The space H1g is compactly embedded in L2

g.

Proof. Assume u ∈ H1g and set v := u

√g. Then

∇v − y

4v =

√g∇u. (47.17)

Fix R > 0. By integration by parts, we have∫BR

g|∇u|2 dy =∫

BR

|∇v|2 dy +116

∫BR

|y|2|v|2 dy − 12

∫BR

vy · ∇v dy

=∫

BR

|∇v|2 dy +116

∫BR

|y|2|v|2 dy

+n

4

∫BR

|v|2 dy − R

4

∫∂BR

v2 dσ.

Since v ∈ L2(Rn) and∫

Rn v2 dy =∫∞0

∫∂Br

v2 dσ dr, there exists a sequence Rj →∞ such that Rj

∫∂BRj

v2 dσ → 0. Therefore, as R = Rj →∞, we obtain

∫Rn

g|∇u|2 dy ≥ 116

∫Rn

|y|2|u|2g dy. (47.18)

Now assume uk → u weakly in H1g . Then Rellich’s theorem guarantees uk → u

in L2loc(R

n). Denoting by ‖ · ‖2,g the norm in L2g, we have

‖uk − u‖22,g =∫|y|≤R

|uk − u|2g dy +∫|y|>R

|uk − u|2g dy =: Ak + Bk,

where Ak → 0 as k →∞ and

Bk ≤ R−2

∫|y|>R

|uk − u|2|y|2g dy ≤ c1R−2‖uk − u‖2H1

g≤ c2R

−2

due to (47.18). These estimates guarantee uk → u in L2g.

Lemma 47.10. For any f ∈ L2g there exists a unique u ∈ H2

g such that Lu = f .

Proof. We shall write∫

f instead of∫

Rn f(y) dy, Lq instead of Lq(Rn), and weset Hk := W k,2(Rn). Denote F (u) := 1

2

∫|∇u|2g −

∫fug. Then F achieves its

minimum in H1g for a unique u satisfying

−∆u− y

2· ∇u = f in D′(Rn).

436 Appendices

Standard regularity results imply u ∈ H2loc. Setting v := u

√g, we have

v ∈ H2loc ∩H1 by (47.17), (47.18). Moreover,

−∆v +(n

4+|y|216

)v = f

√g. (47.19)

Multiplying this equation by (−∆v)φk, where φk(y) := φ0(|y|/k) and φ0 : R+ →[0, 1] is a smooth function satisfying φ0(s) = 1 for s ≤ 1 and φ0(s) = 0 for s ≥ 2,we obtain∫

|∆v|2φk +∫|∇v|2

(n

4+|y|216

)φk

=∫ √

gf(−∆v)φk −18

∫(y · ∇v)vφk −

∫v∇v ·

(n

4+|y|216

)∇φk.

Using Cauchy’s inequality,√

gf, |y|v ∈ L2 and |∇φk| ≤ C/k, we get∫ (12|∆v|2 +

(n

8+|y|216

)|∇v|2

)φk

≤ 12

∫|f |2g + C

∫|y|2|v|2 +

C

k

∫|v||∇v|+ C

∫|yv||∇v||y||∇φk|

=: A + B + Ck + Dk.

(47.20)

We have |y||∇φk(y)| ≤ |y|k

∣∣∇φ0

( |y|k

)∣∣ ≤ C, hence

|yv||∇v||y||∇φk| ≤ C|yv||∇v| ∈ L1.

Since y∇φk → 0 a.e., we get Dk → 0. Obviously Ck → 0, hence, letting k →∞ in(47.20) we deduce∫ (1

2|∆v|2 +

(n

8+|y|216

)|∇v|2

)≤ 1

2

∫|f |2g + C

∫|y|2|v|2 < ∞. (47.21)

In particular, ∆v ∈ L2. Since also v ∈ L2, we have v ∈ H2 by standard ellipticregularity. In addition, (47.19) implies |y|2v ∈ L2 and inequality (47.21) guarantees|y||∇v| ∈ L2. Let us now write

∂2v

∂xi∂xj=√

g∂2u

∂xi∂xj+

yj

4√

g∂u

∂xi+

yi

4√

g∂u

∂xj+

yiyj

16√

gu +δij

4√

gu

=: A1 + A2 + A3 + A4 + A5.

(47.22)

The LHS of (47.22) belongs to L2 since v ∈ H2. Next we have A2, A3 ∈ L2 dueto |y||∇v| ∈ L2, |y|2v ∈ L2 and (47.17). Finally, A4 ∈ L2 due to |y|2v ∈ L2, andA5 ∈ L2. Consequently, A1 ∈ L2, which proves u ∈ H2

g .

We will also need the following lemma.

Lemma 47.11. H1g → Lz

g, where z = 2∗ if n > 2, z ∈ [2,∞) is arbitrary if n ≤ 2.

Proof. First note that H1(Rn) → Lz(Rn). Assume u ∈ H1g . Then (47.17) and

(47.18) imply ∇(u√

g) ∈ L2 and

‖∇(u√

g)‖L2 ≤ C‖u‖H1g,

hence u√

g ∈ Lz and ‖u√g‖Lz ≤ C‖u‖H1g. Now the inequality∫

|u|zg dy ≤∫|u|zgz/2 dy

concludes the proof.

Remarks 47.12. (i) Lemmas 47.9 and 47.11 guarantee that H1g is compactly

embedded in Lp+1g for any p ∈ [1, pS).

(ii) The proofs of Lemmas 47.10 and 47.11 show that H2g → Lz

g, where z =2n/(n− 4) if n > 4, z ∈ [2,∞) is arbitrary if n ≤ 4.

Lemma 47.13. Let λL1 < λL

2 < · · · denote all distinct eigenvalues of L. ThenλL

k = (n + k − 1)/2, k = 1, 2, . . . , and the eigenspaces are

Ker (L− λLk ) = Span Dβφ1 : |β| = k − 1,

where φ1(y) = e−|y|2/4, Dβ = ∂β11 · · · ∂βn

n , |β| = β1 + · · ·+ βn.

Proof. Let u ∈ L2g and let u denote the Fourier transform of u. Since | · |mu ∈

L2(Rn) for any m ≥ 0, we have u ∈⋂

m≥0 Hm(Rn) ⊂ C∞(Rn). Assume Lu = λu.Applying the Fourier transform we obtain

|ξ|2u(ξ) +n

2u(ξ) +

12ξ · ∇u(ξ) = λu(ξ).

Set v(ξ) = e|ξ|2u(ξ). Then v ∈ C∞(Rn) and

ξ · ∇v(ξ) = (2λ− n)v(ξ),

which guarantees that v is a homogeneous function of degree (2λ − n) (cf. theEuler identity for homogeneous functions). As v ∈ C∞(Rn), the degree (2λ − n)has to be a nonnegative integer, hence v = Pk−1, where Pk−1 is a homogeneouspolynomial of degree (k−1) and k ∈ 1, 2, . . .. Then u(ξ) = Pk−1(ξ)e−|ξ|2 , henceu = cPk−1(D)φ1.

438 Appendices

48. Appendix B: Linear parabolic equations

This appendix is devoted to the estimates and various notions of solutions of linearparabolic equations.

48.1. Parabolic regularity

Let Ω be an arbitrary domain in Rn and T > 0. We consider the problem

ut + Au = f in QT , (48.1)

where the operator A is defined in (47.1) and its coefficients aij , bi, c depend onz := (x, t) ∈ QT ,∑

i,j

aij(z)ξiξj ≥ λ|ξ|2 for all z ∈ QT , ξ ∈ Rn. (48.2)

A strong solution of (48.1) is a function u ∈ W 2,1;1loc (QT ) satisfying (48.1) a.e.

The following result (cf. [338, Theorems 7.13, 7.15, 7.17 and Corollary 7.16])contains the basic interior and interior-boundary parabolic Lp-estimates, and anexistence-uniqueness statement. See also [320] for additional results concerningparabolic Lp-theory.

Theorem 48.1. Let Ω be an arbitrary bounded domain in Rn. Assume (48.2) and

(47.3). Let u ∈ W 2,1;ploc ∩Lp(QT ), 1 0, then

‖u‖2,1;p;Q′ ≤ C(‖u‖p;QT + ‖f‖p;QT ), (48.3)

where C depends only on n, p, QT , Q′, λ, Λ, and the moduli of continuity of the aij .(ii) Let Ω be of class C2 and either Σ be an open subset of ST or Σ = PT . Assumeu ∈ W 2,1;p(QT ) and u = 0 on Σ. Let Q′ ⊂ QT , dist(Q′,PT \ Σ) > 0 if Σ = PT .Then (48.3) is true, where C depends also on Σ.(iii) Let Ω be of class C2, ϕ ∈ W 2,1;p(QT ), f ∈ Lp(QT ). Then there exists aunique (strong) solution u of (48.1) satisfying u = ϕ on PT . Moreover, u satisfiesthe estimate

‖u‖2,1;p;QT ≤ C(‖f‖p;QT + ‖ϕ‖2,1;p;QT

).

The following result (cf. [338, Theorems 4.28 and 5.14]) contains the basicinterior-boundary parabolic Schauder estimate and an existence-uniqueness state-ment. We restrict ourselves to global estimates; local estimates can be easily de-rived by applying this theorem to the function uψ where ψ is a smooth cut-offfunction. See also [214] for additional results concerning parabolic Schauder the-ory.

48. Appendix B: Linear parabolic equations 439

Theorem 48.2. Assume (48.2). Let α ∈ (0, 1) and let Ω be a bounded domain ofclass C2+α. Assume aij , bi, c, f ∈ BUCα,α/2(QT ), ϕ ∈ BUC2+α,1+α/2(QT ).

(i) If u ∈ BUC2+α,1+α/2(QT ) is a solution of (48.1) satisfying u = ϕ on PT , then

|u|2+α;QT ≤ C(‖u‖∞ + |f |α;QT + |ϕ|2+α;QT

),

where C depends only on n, α, λ, Ω and the norms of aij , bi, c in BUCα,α/2(QT ).

(ii) There exists a unique solution u ∈ C(QT )∩C2,1(QT ) of (48.1) satisfying u = ϕon PT . If ϕt + Aϕ = f on ∂Ω× 0, then u ∈ BUC2+α,1+α/2(QT ) and

|u|2+α;QT ≤ C(|f |α;QT + |ϕ|2+α;QT

).

Remark 48.3. (i) Assume that A has constant coefficients. Then, by similararguments as in Remark 47.4(i), one can deduce the following regularity resultsfrom Theorems 48.1(i) and 48.2(i). If u, f ∈ Lp

loc(QT ) for some 1 < p < ∞ andut + Au = f in D′(QT ), then u ∈ W 2,1;p

loc (QT ). If u, f ∈ Cα,α/2(QT ) for some0 < α < 1 and ut + Au = f in D′(QT ), then u ∈ C2+α,1+α/2(QT ).

(ii) (Neumann boundary conditions) Under the assumptions Ω bounded,(48.2), (47.3), aij ∈ C(QT ), 1 < p <∞ and f ∈ Lp(QT ), if u ∈ W 2,1;p

loc ∩Lp(QT ) isa strong solution of (48.1) and satisfies ∂νu = 0 on ST and u = 0 on Ω×0, thenwe have the estimate ‖u‖2,1;p;QT ≤ C‖f‖p;QT . Similarly, Theorem 48.2(i) remainsvalid if the condition u = ϕ on PT is replaced by ∂νu = ∂νϕ on ST and u = ϕ onΩ×0. These facts follow from [337, Theorem 7.20] (see also [161, Theorem 8.2])and [337, Theorem 4.31], respectively. For existence-uniqueness results analogousto Theorem 48.2(ii), see [337, Theorem 4.31].

In the rest of Appendix B and in Appendix C we shall restrict ourselves to theLaplace operator for simplicity, but many results can be extended to more generaluniformly elliptic divergence form operators with sufficiently smooth coefficients.

48.2. Heat semigroup, Lp-Lq-estimates, decay, gradientestimates

In this subsection we collect some useful properties of the Dirichlet heat semigroup.Let Ω be an arbitrary domain in R

n and let −A2 denote the Dirichlet Laplacianin L2(Ω), that is the Laplacian on L2(Ω) subject to homogeneous Dirichlet bound-ary conditions (see [154] for its precise definition and for the proof of the followingstatements). Then −A2 is a nonnegative self-adjoint operator and it generates aC0-semigroup e−tA2 on L2(Ω). The space L1 ∩ L∞(Ω) is invariant under e−tA2

and e−tA2 may be extended from L1 ∩L∞(Ω) to a positive contraction semigroupTp(t) on Lp(Ω) for each 1 ≤ p ≤ ∞. These semigroups are strongly continuous if1 ≤ p < ∞ and T∞(t)f → f as t → 0+ in the weak-star topology. In addition,

440 Appendices

Tp(t)f = Tq(t)f for f ∈ Lp ∩ Lq(Ω) and p, q ∈ [1,∞]. If no confusion seems likely,we will denote all the semigroups Tp, 1 ≤ p ≤ ∞, by the same symbol e−tA and callthem the heat semigroup in Ω (more precisely, the Dirichlet heat semigroup inΩ, or the heat semigroup in Ω with homogeneous Dirichlet boundary conditions).Note that u = e−tAf solves the heat equation ut−∆u = 0 in Ω× (0,∞). In addi-tion, if Ω is smooth enough (for instance if it satisfies an exterior cone conditionat each point of ∂Ω), then u ∈ C(Ω× (0,∞)) and u = 0 on ∂Ω× (0,∞).

There exists a positive C∞-function GΩ : Ω × Ω × (0,∞) → R (Dirichletheat kernel) such that (e−tAf)(x) =

∫Ω GΩ(x, y, t)f(y) dy for any f ∈ Lp(Ω),

1 ≤ p ≤ ∞ (the subscript Ω in GΩ will be often omitted if no confusion is likely).In addition,

GΩ1(x, y, t) ≤ GΩ2(x, y, t) (48.4)

whenever Ω1 ⊂ Ω2 and x, y ∈ Ω1, and GΩ(x, y, t) = GΩ(y, x, t) for all x, y ∈ Ω andt > 0. If Ω = R

n, then GRn(x, y, t) = G(x − y, t), where

G(x, t) = Gt(x) := (4πt)−n/2e−x2/4t (48.5)

is the Gaussian heat kernel, hence e−tAf = Gt ∗ f . Note that the functions Gt

satisfy the semigroup property under convolution:

Gt+s = Gt ∗Gs, s, t > 0. (48.6)

Let us also observe that if λ > σ(−A2) and Bλ := (λ + A2)−1, then Bλ =∫ t

0e−λte−tA2 dt and

KΩ,λ(x, y) :=∫ t

0

e−λtGΩ(x, y, t) dt (48.7)

is the kernel of the operator Bλ, that is Bλf(x) =∫Ω KΩ,λ(x, y)f(y) dy. Notice

that for each f ∈ L2(Ω), Bλf is the unique solution of the problem

λu −∆u = f in H−1(Ω), u ∈ H10 (Ω),

and KΩ,λ is the Green function of this problem. If Ω is bounded and if there is norisk of confusion, we denote simply K(x, y) = KΩ(x, y) = KΩ,0(x, y), which is the(elliptic) Green kernel of the Dirichlet Laplacian. Moreover, for n ≥ 3, we have

KΩ(x, y) ≤ Cn|x− y|2−n, (48.8)

as a consequence of (48.4), (48.5) and (48.7).The following Lp-Lq-estimate for the heat semigroup is of fundamental impor-

tance in the study of semilinear problems.

Proposition 48.4. Let (e−tA)t≥0 be the heat semigroup in Rn and Gt(x) =

G(x, t) the Gaussian heat kernel. We have the following properties.(a) ‖Gt‖1 = 1 for all t > 0.(b) If Φ ≥ 0, then e−tAΦ ≥ 0 and ‖e−tAΦ‖1 = ‖Φ‖1.(c) If 1 ≤ q ≤ ∞, then ‖e−tAΦ‖q ≤ ‖Φ‖q for all t > 0.

(d) If 1 ≤ p < q ≤ ∞ and 1/r = 1/p − 1/q, then ‖e−tAΦ‖q ≤ (4πt)−n/(2r)‖Φ‖p

for all t > 0.(e) For an arbitrary domain Ω ⊂ R

n, assertions (c) and (d) remain valid if e−tA

is replaced with the Dirichlet heat semigroup in Ω.

Proof. Statement (a) is well known, statement (b) follows from Fubini’s theoremand part (a). Statement (c) follows from the contractivity of the semigroup Tq(t)(see above); it also easily follows from the estimate ‖Gt ∗ Φ‖q ≤ ‖Gt‖1‖Φ‖q.

Interpolating between (b) and the inequality ‖e−tAΦ‖∞ ≤ (4πt)−n/2‖Φ‖1 weobtain

‖e−tAΦ‖q ≤ (4πt)−(n/2)(1−1/q)‖Φ‖1. (48.9)

Interpolating between (48.9) and (c) yields (d).To prove assertion (e), denote by e−tAΩ the Dirichlet heat semigroup in Ω. Let

Φ(x) = Φ(x) if x ∈ Ω, Φ(x) = 0 otherwise. By (48.4) we have

|e−tAΩΦ| ≤ e−tAΩ |Φ| ≤ e−tA|Φ|.

The conclusion follows from assertions (c) and (d).

In the case of bounded domains, we have the following classical property ofuniform exponential decay.

Proposition 48.5. Let Ω be an arbitrary bounded domain and let (e−tA)t≥0 bethe Dirichlet heat semigroup in Ω. For all 1 ≤ p ≤ ∞ and all Φ ∈ Lp(Ω), thereholds

‖e−tAΦ‖p ≤ C(Ω)e−λ1t‖Φ‖p, t ≥ 0. (48.10)

Proof. If 0 < t < 2, then (48.10) follows from Proposition 48.4(c). We may thusassume t ≥ 2. It is well known that

‖e−tAΦ‖2 ≤ e−λ1t‖Φ‖2, t ≥ 0. (48.11)

Using (48.11), Proposition 48.4(d) for p = 2, q = ∞, Holder’s inequality and|Ω|1/p ≤ max(1, |Ω|), we get

‖e−tAΦ‖p ≤ |Ω|1/p‖e−tAΦ‖∞ ≤ (4π)−n/4|Ω|1/p‖e−(t−1)AΦ‖2≤ C(Ω)e−λ1(t−2)‖e−AΦ‖2.

442 Appendices

The assertion then follows from ‖e−AΦ‖2 ≤ ‖Φ‖2 ≤ C(Ω)‖Φ‖p if p ≥ 2, and from‖e−AΦ‖2 ≤ ‖Φ‖p (owing to Proposition 48.4(d)) if p < 2.

In the case of the whole space and integrable initial data, the asymptotic be-havior is described by a multiple of the Gaussian heat kernel (see [175], or [168]for further results).

Proposition 48.6. Let Φ ∈ L1(Rn) and put M =∫

Rn Φ dx.(i) There holds

‖e−tAΦ−M Gt‖1 → 0, t→∞.

(ii) If, in addition, xΦ(x) ∈ L1(Rn), then

‖e−tAΦ−M Gt‖1 ≤ Ct−1/2‖xΦ(x)‖1, t > 0,

where C = C(n) > 0.

Proof. We first establish assertion (ii). Let Φ ∈ L1(R

n; (1 + |x|) dx).(

e−tAΦ−M Gt

)(x) = (4πt)−n/2

∫Rn

(e−|x−y|2/4t − e−|x|2/4t

)Φ(y) dy

=(4πt)−n/2

2√

t

∫Rn

∫ 1

0

y · (x− θy)√t

e−|x−θy|2/4tΦ(y) dy dθ.

Using sups>0 se−s2/8 < ∞ and Fubini’s theorem, we deduce that

‖e−tAΦ−M Gt‖1 ≤ Ct−(n+1)/2

∫ 1

0

∫Rn

∫Rn

e−|x−θy|2/8t|y||Φ(y)| dx dy dθ

= Ct−1/2

∫ 1

0

∫Rn

|y||Φ(y)| dy dθ = Ct−1/2‖xΦ(x)‖1.

Let us next prove assertion (i). Fix Φ ∈ L1(Rn) and pick a sequence ϕj ∈D(Rn) such that

∫Rn ϕj dx = M and ϕj → Φ in L1(Rn). For each j we write

‖e−tAΦ−M Gt‖1 ≤ ‖e−tAϕj −M Gt‖1 + ‖e−tA(Φ− ϕj)‖1≤ ‖e−tAϕj −M Gt‖1 + ‖Φ− ϕj‖1.

By assertion (ii), it follows that

lim supt→∞

‖e−tAΦ−M Gt‖1 ≤ ‖Φ− ϕj‖1

and the conclusion follows by letting j →∞.

We conclude with a smoothing estimate for the gradient, which we state withoutproof (this follows from [320, Theorem IV.16.3, p. 413]).

Proposition 48.7. Let Ω be a domain of class C2+α for some α ∈ (0, 1) and let(e−tA)t≥0 be the Dirichlet heat semigroup in Ω. For all Φ ∈ L∞(Ω), there holds

‖∇e−tAΦ‖∞ ≤ C(Ω)(1 + t−1/2)‖Φ‖∞, t > 0.

48.3. Weak and integral solutions

In this subsection we compare various notions of solutions of the inhomogeneouslinear heat equation. Related semigroup and smoothing properties will be de-scribed in Appendix C (Subsection 49.2).

Assume that Ω is a bounded domain of class C2+α for some α ∈ (0, 1). Similarlyas in Remarks 15.4(iv) and (v), we may define integral and weak L1

δ-solutions ofthe linear problem

ut −∆u = f, x ∈ Ω, t ∈ (0, T ),

u = 0, x ∈ ∂Ω, t ∈ (0, T ),

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (48.12)

as follows.

Definition 48.8. (i) Let u0 ∈ L1δ(Ω) and f ∈ L1

loc((0, T ), L1δ(Ω)). A function

u ∈ C([0, T ), L1δ(Ω)) ∩ L1

loc((0, T ), L1(Ω)) is a weak L1δ-solution of (48.12) if

u(·, 0) = u0 and, for any 0 < τ < t < T ,∫ t

τ

∫Ω

fϕ = −∫ t

τ

∫Ω

u(ϕt + ∆ϕ) −∫

Ω

u(τ)ϕ(τ)

for all ϕ ∈ C2(Ω× [τ, t]) such that ϕ = 0 on ∂Ω× [τ, t] and ϕ(t) = 0.(48.13)

(ii) Let u0, f be nonnegative measurable functions and let G denote the Dirichletheat kernel in Ω. Then

u(x, t) :=∫

Ω

G(x, y, t)u0(y) dy +∫ t

0

∫Ω

G(x, y, t− s)f(y, s) dy ds ≤ ∞

is called the integral solution of (48.12).

Proposition 48.9. Let Ω be as above and let u0 ∈ L1δ(Ω).

(i) If f ∈ L1loc([0, T ), L1

δ(Ω)), then problem (48.12) possesses a unique weak L1δ-

solution. Moreover u ∈ L1loc([0, T ), L1(Ω)) and (48.13) is also satisfied for τ = 0.

(ii) If f ∈ L1loc((0, T ), L1

δ(Ω)) and problem (48.12) possesses a weak L1δ-solution,

then f ∈ L1loc([0, T ), L1

δ(Ω)).

Proof. (i) Let f ∈ L1loc([0, T ), L1

δ(Ω)). We first prove the uniqueness. Assumethat u1, u2 are two weak solutions of (48.12) and set w := u1 − u2. Then w is aweak solution of the homogeneous problem (48.12) (with f = 0 and u0 = 0). Inparticular, ∫ t

τ

∫Ω

w(ϕt + ∆ϕ) +∫

Ω

w(τ)ϕ(τ) = 0 (48.14)

444 Appendices

whenever 0 < τ < t < T and ϕ ∈ C2,1(Ω× [τ, t]) satisfies ϕ = 0 on ∂Ω× [τ, t] andϕ(t) = 0.

Fix t ∈ (0, T ). Let ψ ∈ D(Qt) and let ϕ = ϕψ be the solution of the problem

−ϕt −∆ϕ = ψ in Qt,

ϕ = 0 on St,

ϕ(t) = 0 in Ω.

Then passing to the limit in (48.14) as τ → 0 we obtain∫ t

0

∫Ω

wψ = 0,

hence w = 0 a.e.In order to prove the existence, we may assume that u0 ≥ 0 and f ≥ 0 (otherwise

we decompose u0 and f into their positive and negative parts and use the linearityof the problem (48.12)). Set u0,k := min(u0, k) and fk := min(f, k), k = 1, 2, . . . .Let uk be the (strong) solution of (48.12) with f and u0 replaced by fk and u0,k,respectively. Then

uk(x, t) =∫

Ω

G(x, y, t)u0,k(y) dy +∫ t

0

∫Ω

G(x, y, t− s)fk(y, s) dy ds, (48.15)

where G denotes the Dirichlet heat kernel in Ω. Passing to the limit in (48.15) ask →∞ we get uk(x, t) u(x, t), where u satisfies

u(x, t) =∫

Ω

G(x, y, t)u0(y) dy +∫ t

0

∫Ω

G(x, y, t− s)f(y, s) dy ds. (48.16)

Notice also that uk ∈ C([0, T ), Lr(Ω)) ∩ W 2,1;rloc (Ω × (0, T )) for all r ∈ (1,∞)

(see Theorem 48.1 and Appendix E). Let 0 ≤ τ < t < T , q > 1 and ϕ ∈C([τ, t], Lq(Ω)) ∩W 2,1;q

loc (Ω × (τ, t)) satisfy ϕ = 0 on ∂Ω × (τ, t) and (ϕt + ∆ϕ) ∈L1(Ω × (τ, t)). Multiplying the equation for uk by ϕ, integrating over Ω × (τ ′, t′)with τ < τ ′ < t′ < t and letting τ ′ → τ , t′ → t, we obtain∫ t

τ

∫Ω

fkϕ = −∫ t

τ

∫Ω

uk(ϕt + ∆ϕ) +∫

Ω

uk(t)ϕ(t) −∫

Ω

uk(τ)ϕ(τ). (48.17)

Set ϕ := ψ, where ψ is the solution of the problem

−ψt −∆ψ = 1 in Qt,

ψ = 0 on St,

ψ(t) = 0 in Ω.

Then (48.17) with τ = 0 implies∫ t

0

∫Ω

uk =∫ t

0

∫Ω

fkψ +∫

Ω

u0,kψ(0) ≤ C(t) <∞,

hence the sequence uk is bounded in L1(Qt), uk → u in L1(Qt) and u ∈L1

loc([0, T ), L1(Ω)).

Next set ϕ := χ, where χ(x, s) = eλ1(s−t)ϕ1(x), which satisfies χs + ∆χ = 0 inQt. For k ≥ j, it follows from (48.17) with τ = 0 and (1.4) that

c1

∫Ω

(uk − uj)(t)δ ≤∫

Ω

(uk − uj)(t)ϕ1 =∫ t

0

∫Ω

(fk − fj)χ +∫

Ω

(u0,k − u0,j)χ(0)

≤ c2

∫ t

0

∫Ω

(fk − fj)δ + c2

∫Ω

(u0,k − u0,j)δ.

This estimate guarantees that uk is a Cauchy sequence in C([0, t], L1δ(Ω)), hence

u ∈ C([0, T ), L1δ(Ω)).

Finally, fix 0 ≤ τ < t < T and ϕ ∈ C2(Ω× [τ, t]) satisfying ϕ = 0 on ∂Ω× [τ, t]and ϕ(t) = 0. Then passing to the limit in (48.17) as k → ∞ we see that u is aweak solution of (48.12) and that (48.13) is also satisfied for τ = 0.

For future reference, we note that the solution u that we have just constructedsatisfies ∫ t

0

∫Ω

|f |δ ≤ C

∫Ω

|u(t)|δ, 0 < t < T, (48.18)

where C remains bounded for T bounded. Indeed (still assuming u0, f ≥ 0 withoutloss of generality), (48.18) follows by passing to the limit k → ∞ in (48.17) withτ = 0 and ϕ = χ.

(ii) Now assume that problem (48.12) possesses a weak L1δ-solution u. Then, for

each τ ∈ (0, T ), u coincides with the weak L1δ-solution of (48.12) on (τ, T ) with

initial data u(τ), given by part (i). For each t ∈ (0, T ), estimate (48.18) guaranteesthat ∫ t

τ

∫Ω

|f |δ ≤ C

∫Ω

|u(t)|δ, 0 < τ < t,

and the assertion follows by letting τ → 0.

Corollary 48.10. Let Ω be as above, u0 ∈ L1δ(Ω), u ∈ L1

loc(QT ), and f : QT → R

be measurable. Assume that u0, u, f ≥ 0.(i) If f ∈ L1

loc((0, T ), L1δ(Ω)) and u is a weak L1

δ-solution of (48.12), then it is anintegral solution of (48.12).(ii) If u is an integral solution of (48.12), then f ∈ L1

loc((0, T ), L1δ(Ω)) and u is a

weak L1δ-solution of (48.12).

446 Appendices

Proof. If u is a weak solution, then f ∈ L1loc([0, T ), L1

δ(Ω)) by Proposition 48.9(ii),and the proof of Proposition 48.9(i) (cf. formula (48.16)) shows that u is an integralsolution.

Let u be an integral solution of (48.12). Again, the proof of Proposition 48.9(i)guarantees that u is a weak solution provided we show f ∈ L1

loc([0, T ), L1δ(Ω)).

Let fk, uk be as in the proof of Proposition 48.9(i). Let 0 < t < T ′ < T andψ ∈ D(QT ), ψ ≥ 0, ψ(·, t) ≡ 0. Let ϕ be the solution of the problem

−ϕt −∆ϕ = ψ in QT ′ ,

ϕ = 0 on ST ′ ,

ϕ(T ′) = 0 in Ω.

Then there exists ε > 0 such that

ϕ(x, s) ≥ εδ(x) for all (x, s) ∈ Qt.

Multiplying the equation ∂tuk −∆uk = fk by ϕ we obtain

ε

∫ t

0

∫Ω

fkδ ≤∫ T ′

0

∫Ω

fkϕ =∫ T ′

0

∫Ω

ukψ −∫

Ω

u0,kϕ(0) ≤∫ T ′

0

∫Ω

uψ < ∞,

hence∫ t

0

∫Ω fδ <∞, which guarantees f ∈ L1

loc([0, T ), L1δ(Ω)).

Corollary 48.11. Let Ω be as above, q ≥ 1 and let u be a mild Lq-solution of(48.12) (that is u ∈ C([0, T ), Lq(Ω)), u(0) = u0, f ∈ L1

loc((0, T ), L1(Ω)) and (15.5)is true with f(u) replaced by f). Then u is a weak L1

δ-solution of (48.12).

Proof. Fix τ0 ∈ (0, T ) and set u0,1 := u+(τ0), u0,2 := u−(τ0), f1 := f+, f2 := f−,

vi(t) = vi(t; τ0) := e−(t−τ0)Au0,i +∫ t

τ0

e−(t−s)Afi(s) ds, τ0 ≤ t < T, i = 1, 2.

Then vi, i = 1, 2, are nonnegative integral solutions of problem (48.12) with[0, T ), u0, f replaced by [τ0, T ), u0,i, fi. Consequently, vi, i = 1, 2, are weak so-lutions of those problems and v := v1 − v2 is a weak solution of (48.12) on [τ0, T )with initial data u(τ0). On the other hand, v1(t; τ0) − v2(t; τ0) = u(t) for anyt ∈ (τ0, T ), hence u = v is a weak solution of (48.12) on [0, T ).

Remark 48.12. In the case of Ω = Rn, for instance, and of nonnegative data

u0, f , one can also study the relations between local classical nonnegative solutionsof (48.12) and integral solutions. Let Ω = R

n, u0 ∈ L1loc(R

n), f be locally Holdercontinuous in QT , with u0 ≥ 0 a.e. and f ≥ 0. Assume that 0 ≤ u ∈ C2,1(QT ) ∩C([0, T ); L1

loc(Rn)) is a solution of ut − ∆u = f in QT , with u(·, 0) = u0. Then

u satisfies (48.16) in QT , where all the integrals are in particular finite (see [482]and cf. also [526]). Such property may be useful, e.g., when considering problemsof Fujita-type.

49. Appendix C: Linear theory in Lpδ-spaces and in uniformly local spaces 447

49. Appendix C: Linear theory in Lpδ-spaces and in

uniformly local spaces

In this section, we state and prove some useful properties of the Laplace and heatequations in weighted Lebesgue spaces Lp

δ(Ω) and in uniformly local Lebesguespaces Lp

ul(Rn). We refer to Section 1 for the definition of these spaces.

49.1. The Laplace equation in Lpδ-spaces

Very weak, or L1δ, solutions of the Laplace equation (47.8) have been introduced in

Definition 3.1. We have the following existence-uniqueness result (see [94]; estimate(49.1) is proved there for q = 1 and in the general case in [107]).

Theorem 49.1. Let Ω ⊂ Rn be a bounded domain of class C2+α for some α ∈

(0, 1), and let f ∈ L1δ(Ω). Then there exists a unique u ∈ L1(Ω) such that u is

an L1δ-solution of problem (47.8). Moreover, for all 1 ≤ q < n/(n − 1), we have

u ∈ Lq(Ω) and‖u‖q ≤ C(n, q, Ω) ‖f‖1,δ. (49.1)

Furthermore, the maximum principle is satisfied, i.e.: f ≥ 0 a.e. implies u ≥ 0a.e.

Proof. We start by proving the uniqueness. Thus assume that u ∈ L1(Ω) is anL1

δ-solution of problem (47.8) with f = 0. Take any h ∈ D(Ω) and let ϕ ∈ C2(Ω)be the classical solution of

−∆ϕ = h in Ω,

ϕ = 0 on ∂Ω.

Then

∫Ω uh dx = 0 by (3.3). It follows that u = 0, hence the uniqueness assertion.

Let us show the existence. We may assume f ≥ 0 without loss of generality(writing f = f+ − f−). Let fi = min(f, i) and denote by ui the strong solution of(47.8) with f replaced by fi. Let Θ be the classical solution of (19.27). For j ≥ i,we have fj ≥ fi ≥ 0, hence uj ≥ ui ≥ 0 by the maximum principle. Testing theequation for uj − ui with Θ, we have

‖uj − ui‖1 =∫

Ω

(uj − ui) dx =∫

Ω

(fj − fi)Θ dx.

Since fi → f in L1δ(Ω), we deduce that ui is a Cauchy sequence in L1(Ω), and

we denote by u ∈ L1(Ω) its limit. Observe that u ≥ 0. For any ϕ ∈ C2(Ω) withϕ = 0 on ∂Ω, we then have∫

Ω

u(−∆ϕ) dx = limi→∞

∫Ω

ui(−∆ϕ) dx = limi→∞

∫Ω

fiϕdx =∫

Ω

fϕdx, (49.2)

448 Appendices

hence u is an L1δ-solution of (47.8).

Next, the choice ϕ = Θ in (49.2) yields∫Ω u dx =

∫Ω fΘ dx. Assuming again

f ≥ 0, this implies estimate (49.1) for q = 1. The case 1 < q < n/(n − 1) will beproved along with Theorem 49.2.

The following results describe the optimal regularity of the Dirichlet Laplacianin the scale of Lp

δ-spaces (see [84], [200] for Theorem 49.2 and [489] for Theo-rem 49.3). The proofs will be given in Subsection 49.4 below.


(0, 1). Assume that 1 ≤ p ≤ q ≤ ∞ satisfy

1p− 1

q<

2n + 1

. (49.3)

Let f ∈ L1δ(Ω) and let u be the L1

δ-solution of (47.8).(i) If f ∈ Lp

δ(Ω), then u ∈ Lqδ(Ω) and

‖u‖q,δ ≤ C(p, q, Ω) ‖f‖p,δ. (49.4)

(ii) If f+ ∈ Lpδ(Ω), then u+ ∈ Lq

δ(Ω) and

‖u+‖q,δ ≤ C(p, q, Ω) ‖f+‖p,δ. (49.5)


(0, 1). Assume that 1 ≤ p < q ≤ ∞ satisfy

1p− 1

q>

2n + 1

.

Then there exists f ∈ Lpδ(Ω) such that the L1

δ-solution u of (47.8) satisfies

u ∈ Lqδ(Ω).

Remarks 49.4. (a) In Theorem 49.2 one may take in particular q = ∞ forp > (n + 1)/2 and any q < (n + 1)/(n− 1) for p = 1.

(b) By a density argument, it is easy to see that the L1δ-solution u of (47.8) is

given by u(x) =∫Ω K(x, y)f(y) dy, where K(x, y) is the Dirichlet Green kernel in

Ω.(c) By similar arguments as in the proofs of Theorems 49.2 and 49.3 (see [489]

for details), one can obtain further optimal regularity properties of the solution uof (47.8). Namely, assuming 1 ≤ p ≤ q ≤ ∞:

• if n+1p − n

q < 2, then u ∈ Lq(Ω) and ‖u‖q ≤ C‖f‖p,δ;

• if n+1p − n

q > 2, then there exists f ∈ Lpδ such that u ∈ Lq(Ω);

• if n+1p − n

q < 1, then u ∈W 1,q0 (Ω) and ‖u‖1,q ≤ C‖f‖p,δ;

• if n+1p − n

q > 1, then there exists f ∈ Lpδ such that u ∈ W 1,q

0 (Ω).

In particular, it follows that if f ∈ Lpδ for some p > 1, then u ∈ W 1,q

0 (Ω) for q > 1close to 1, so that the boundary conditions in (47.8) are also satisfied in the senseof traces. We note that in the example constructed in the proof of Theorem 49.3,the solution u possesses a singularity at a (single) boundary point a ∈ ∂Ω andthat u ∈W 2,m

loc (Ω \ a) for all finite m.

(d) Theorem 49.2 remains true in case of equality in (49.3) provided p > 1,q < ∞ and n = 2 (see [362], where equality cases in Remark (c) are also treated).

(e) If f ∈ L1δ, then, for each α ∈ (0, 1), we have u/δα ∈ L1(Ω) and ‖u/δα‖1 ≤

C(α)‖f‖1,δ. This can be shown by using the singular test-function ξ from Lem-ma 10.4 for smooth f and the general case follows by density.

We close this subsection by proving a useful, simple consequence of Theo-rem 49.2 (cf. [449, Proposition 2.3]).

Proposition 49.5. Let Ω ⊂ Rn be a bounded domain of class C2+α for some

α ∈ (0, 1). Let f ∈ L1δ(Ω) and let u be the L1

δ-solution of (47.8). Then, for any1 ≤ k < (n + 1)/(n− 1), we have

‖u‖k,δ ≤ C(Ω, k)(‖u+‖1,δ + ‖f−‖1,δ

).

Proof. Using (3.3) with ϕ = ϕ1 and (1.4), we obtain

∫Ω

|f |ϕ1 =∫

Ω

fϕ1 + 2∫

Ω

(f−)ϕ1

= λ1

∫Ω

uϕ1 + 2∫

Ω

(f−)ϕ1 ≤ C(Ω)(‖u+‖1,δ + ‖f−‖1,δ

).

Applying Theorem 49.2(i) with p = 1 and using ϕ1 ≥ c1δ, we deduce that

‖u‖k,δ ≤ C(Ω, k)‖f‖1,δ ≤ C(Ω, k)(‖u+‖1,δ + ‖f−‖1,δ

).

450 Appendices

49.2. The heat semigroup in Lpδ-spaces

We start by introducing a natural extension of the Dirichlet heat semigroup. Herewe also use the spaces Lp

ϕ1(Ω), which are defined similarly as Lp

δ(Ω). Note that ifΩ is C2-smooth, then Lp

ϕ1(Ω) .= Lp

δ(Ω), due to (1.4).

Proposition and Definition 49.6. Let Ω be an arbitrary bounded domain in Rn.

The Dirichlet heat semigroup admits a unique extension to L1ϕ1

(Ω), still denotedby (e−tA)t≥0. It is a contraction semigroup on L1

ϕ1(Ω), which satisfies

‖e−tAφ‖1,ϕ1 = e−λ1t‖φ‖1,ϕ1 , t ≥ 0, φ ∈ L1ϕ1

(Ω). (49.6)

Moreover the maximum principle is satisfied, i.e.:

φ ∈ L1ϕ1

(Ω) and φ ≥ 0 a.e. imply e−tAφ ≥ 0 a.e. (49.7)

Furthermore, for each 1 < p < ∞, (e−tA)t≥0 restricts to a contraction semigroupon Lp

ϕ1(Ω), which satisfies

‖e−tAφ‖p,ϕ1 ≤ e−(λ1/p)t‖φ‖p,ϕ1 , t ≥ 0, φ ∈ Lpϕ1

(Ω). (49.8)

In addition, if Ω is of class C2, then we have

‖e−tAφ‖p,δ ≤ C(Ω) e−(λ1/p)t‖φ‖p,δ, t ≥ 0, φ ∈ Lpδ(Ω), 1 ≤ p <∞. (49.9)

Proof. Let φ ∈ L2(Ω) with φ ≥ 0. Since e−tA is self-adjoint on L2(Ω) ande−tAϕ1 = e−λ1tϕ1, we have, for all t ≥ 0,

‖e−tAφ‖1,ϕ1 = (e−tAφ, ϕ1) = (φ, e−tAϕ1) = e−λ1t(φ, ϕ1) = e−λ1t‖φ‖1,ϕ1 .(49.10)

Writing φ = φ+−φ− and using the linearity and the positivity preserving propertyof e−tA, it follows that (49.10) is true for all φ ∈ L2(Ω).

Now fix φ ∈ L1ϕ1

(Ω) and pick a sequence φi in L2(Ω), such that φi → φ inL1

ϕ1(Ω). For each fixed t > 0, (49.10) implies

‖e−tAφi − e−tAφj‖1,ϕ1 = e−λ1t‖φi − φj‖1,ϕ1 ,

thus e−tAφi is a Cauchy sequence in L1ϕ1

(Ω). Consequently, we may definee−tAφ := limi→∞ e−tAφi, and it follows from (49.10) that the limit is indepen-dent of the choice of the sequence φi, hence the uniqueness assertion. Moreover,(49.6) is satisfied. On the other hand, if φ ≥ 0, by choosing φi = min(φ, i) ≥ 0, weobtain (49.7).

Finally let 1 ≤ p < ∞ and φ ∈ Lpϕ1

(Ω). By using Jensen’s inequality and∫Ω

G(x, y, t) dy ≤ 1, we have

|e−tAφ|p ≤ e−tA(|φ|p), φ ∈ Lpϕ1

(Ω) (49.11)

(first assume φ ∈ Lp(Ω) and then argue by density). Therefore, using (49.6), weget

‖e−tAφ‖pp,ϕ1

= ‖|e−tAφ|p‖1,ϕ1 ≤ ‖e−tA(|φ|p)‖1,ϕ1

= e−λ1t‖|φ|p‖1,ϕ1 = e−λ1t‖φ‖pp,ϕ1

,

hence (49.8). If Ω is smooth, then (49.9) follows from (49.8) and (1.4).

The following result provides optimal smoothing estimates for the Dirichlet heatsemigroup in the scale of Lp

δ-spaces (see [200] for assertion (i) and [200], [489] forassertion (ii)). Its proof is postponed to Subsection 49.4 below.

Theorem 49.7. Let Ω ⊂ Rn be a bounded domain of class C2, let 1 ≤ p ≤ q ≤ ∞

and set β = n+12 ( 1

p −1q ).

(i) For all φ ∈ Lpδ(Ω), we have

‖e−tAφ‖q,δ ≤ C(p, q, Ω) ‖φ‖p,δt−β, t > 0. (49.12)

(ii) For all ε > 0, there exist a function φ ∈ Lpδ(Ω) and a constant C > 0, such

that‖e−tAφ‖q,δ ≥ Ct−β+ε, for t > 0 small.

Remarks 49.8. (a) The elliptic and parabolic estimates in Theorems 49.2 and49.7 exhibit a remarkable dimension shift phenomenon: they are similar to those instandard Lp-spaces in n+1 dimensions (cf. Proposition 47.5 and Proposition 48.4).

(b) Assume Ω smooth. Recalling that L∞δ = L∞ and interpolating between

(49.9) with p = 1 and (48.10) with p =∞, we see that there exists C = C(Ω) > 0such that

‖e−tAφ‖p,δ ≤ C e−λ1t‖φ‖p,δ, t ≥ 0, φ ∈ Lpδ(Ω), 1 ≤ p ≤ ∞, (49.13)

which is an alternative to (49.8).

(c) Assume Ω smooth and let u0 ∈ L1δ(Ω). By a density argument, it is easy to see

that u(t) := e−tAu0 satisfies u(x, t) =∫Ω

G(x, y, t)u0(y) dy in Ω×(0,∞). Moreoveru is a weak L1

δ-solution of (48.12) with f = 0 in the sense of Remark 15.4(v).

452 Appendices

49.3. Some pointwise boundary estimates for the heatequation

We here state and prove some pointwise estimates for the heat (and the Laplace)equation, involving the distance to the boundary, which are essential to establishthe Lp

δ-properties stated above. Some of them are also used at other places.

Proposition 49.9. Let Ω be a bounded domain of class C2. There exists C =C(Ω) > 0 such that, for all φ ∈ L∞(Ω),

∣∣(e−tAφ)(x)∣∣ ≤ C‖φ‖∞

δ(x)√t

, x ∈ Ω, t > 0. (49.14)

Proposition 49.9 can be derived as a consequence of Gaussian estimates for thegradient of the heat kernel [477] (or of the reverse of estimate (49.17) below).However, we shall give a maximum principle based, self-contained proof relying onarguments from [346], [347].

Proof. Step 1. We consider the auxiliary problem

Vt −∆V = 1, x ∈ Ω, t > 0,

V = 0, x ∈ ∂Ω, t > 0,

V (x, 0) = 0, x ∈ Ω,

⎫⎪⎬⎪⎭ (49.15)

and we claim that, for some T = T (Ω) > 0, there holds

V (x, t) ≤ 2√

t δ(x), x ∈ Ω, 0 < t ≤ T. (49.16)

To show (49.16) we use a barrier argument based on the construction of a suit-able supersolution. Fix x1 ∈ Ω and pick x2 ∈ ∂Ω such that δ(x1) =|x1 − x2|.Since the domain Ω is C2-smooth and bounded, one can find 0 < ρ < R indepen-dent of x1, and a ∈ R

n, such that Ω ⊂ D := x ∈ Rn : ρ < |x − a| < R and

Ω ∩B(a, ρ) = x2. For (x, t) ∈ Q := D × (0,∞), we consider the function

V (x, t) = t ϕ(y),

where

y =|x− a| − ρ√

tand ϕ(y) =

y(2− y)

1

if 0 ≤ y < 1,

if y ≥ 1.

The function V is C1 in t on Q and C2 in x on Q := (x, t) ∈ Q : y = 1. Wecompute

V t = ϕ(y)− y

2ϕ′(y) ≥ χy≥1 −

12χ0≤y<1

and, for all (x, t) ∈ Q,

−∆V = −ϕ′′(y)− n− 1|x− a|

√t ϕ′(y) ≥ 2χ0≤y<1 − 2(n− 1)ρ−1

√T χ0≤y<1.

Taking T = (ρ/4(n−1))2, we obtain V t−∆V ≥ 1 in Q. On the other hand, it is easyto see that 0 ≤ V ∈ C(D × (0,∞)), V ∈ C1((0,∞), L2(D)) and V (·, t) ∈ H2(D)for each t > 0. Moreover, V (·, t) → 0 in L∞(D) as t → 0. Consequently, V isa (weak) supersolution to (49.15). It follows from the maximum principle thatV ≤ V in Ω× (0, T ], hence in particular

V (x1, t) ≤ t ϕ((|x1 − a| − ρ)/

√t)≤ 2

√t (|x1 − a| − ρ) = 2

√t δ(x1), 0 < t ≤ T.

Step 2. Let U(t) = e−tAχΩ. For each τ > 0, the maximum principle yieldsU(0)−U(τ) = χΩ− e−τAχΩ ≥ 0, hence U(t)−U(t+ τ) = e−tA(U(0)−U(τ)) ≥ 0.Therefore U is nonincreasing in time. By the variation-of-constants formula, itfollows that

V (t) =∫ t

0

U(s) ds ≥ t U(t).

This combined with (49.16) yields

(e−tAχΩ

)(x) ≤ 2δ(x)√

t, x ∈ Ω, 0 < t ≤ T.

By the maximum principle, we deduce that (49.14) is true for 0 < t ≤ T . If t ≥ T ,using (48.10) with p =∞, we obtain

(e−tAχΩ

)(x) ≤ 2δ(x)√

T‖e−(t−T )AχΩ‖∞ ≤ 2δ(x)√

TCe−λ1(t−T ), x ∈ Ω.

The proposition follows.

Proposition 49.10. Let Ω be an arbitrary domain in Rn. There exist constants

c1 > 0 and c2 ≥ 2 depending only on n, such that the Dirichlet heat kernel G(x, y, t)in Ω satisfies

G(x, y, t) ≥ c1t−n/2,

for all t > 0 and x, y ∈ Ω such that

δ(x) ≥ c2

√t and |x− y| ≤

√t.

Proposition 49.10 is a consequence of the sharp estimate [545]

G(x, y, t) ≥ C1 min(1, δ(x)δ(y)

t

)t−n/2e−C2|x−y|2/t, for t > 0 small, (49.17)

454 Appendices

but the proof of (49.17) is much more delicate. (Estimate (49.17) is in fact provedin [545] for C2 bounded domains and n ≥ 3; the reverse inequality, with differentconstants C1, C2, is also true [153].) Here we give an elementary and self-containedproof of Proposition 49.10 based only on the maximum principle.

Proof. Fix y ∈ Ω, let ρ = δ(y), B = B(y, ρ), and denote

u(x, t) = (4πt)−n/2e−|x−y|2/4t, x ∈ B, t > 0.

For x ∈ ∂B, we have u(x, t) = ρ−ng(tρ−2), where g(s) = (4πs)−n/2e−1/4s. Leta(n) := sups>0 g(s) (which is finite) and put

u(x, t) = u(x, t)−M, M := a(n)ρ−n.

Then u satisfies

ut −∆u = 0, x ∈ B, t > 0,

u ≤ 0, x ∈ ∂B, t > 0,

(49.18)

and moreover u(·, t) → δy −M in the sense of measures, as t→ 0, where δy is theDirac measure at point y. It follows from the maximum principle that G(x, y, t) ≥u(x, t) in B× (0,∞). (More precisely, one can easily show that inequality (52.16),with u(x, t) replaced by G(x, y, t) − u(x, t), is satisfied for f = 0 and u0 = 0; sothe assertion follows from Proposition 52.13(ii).) In particular, if δ(x) ≥ c2

√t and

|x− y| ≤√

t, hence ρ = δ(y) ≥ (c2 − 1)√

t, we obtain

G(x, y, t) ≥((4π)−n/2e−1/4 − a(n)(c2 − 1)−n

)t−n/2 ≥ c1(n)t−n/2

provided we choose c2 = c2(n) > 1 large enough.

Proposition 49.11. Let Ω be a bounded domain of class C2.(i) The Dirichlet heat kernel G(x, y, t) in Ω satisfies

G(x, y, t) ≥ c(t, Ω) δ(x)δ(y), x, y ∈ Ω, t > 0

where the constant c(t, Ω) is uniformly positive for t bounded and bounded awayfrom 0.(ii) There exists a constant c = c(Ω) > 0 such that the Dirichlet Green kernelK(x, y) in Ω satisfies

K(x, y) ≥ c δ(x)δ(y), x, y ∈ Ω.


Remarks 49.12. (i) Proposition 49.11 provides quantitative versions of Hopf’slemma. Namely, let f ∈ L1

δ(Ω) satisfy f ≥ 0 a.e. Then the L1δ(Ω) solution u of the

Laplace equation (47.8) satisfies

u(x) =∫

Ω

K(x, y)f(y) dy ≥ c(Ω) ‖f‖1,δ δ(x), x ∈ Ω.

Likewise, for the heat equation, we have

(e−tAf

)(x) =

∫Ω

G(x, y, t)f(y) dy ≥ c(t, Ω) ‖f‖1,δ δ(x), x ∈ Ω, t > 0. (49.19)

(ii) Estimate (49.19) is sharp in the sense that, for all f ∈ L1δ(Ω),

|(e−tAf)(x)| ≤ c(Ω) t−(n+2)/2 ‖f‖1,δ δ(x), x ∈ Ω, t > 0.

This follows by writing e−tAf = e−(t/2)A(e−(t/2)Af) and combining (49.14) with(49.12) for p = 1, q = ∞.

Again, Proposition 49.11 is a consequence of estimate (49.17). We give a simpleproof essentially based on [345] (see also [92]).

Proof of Proposition 49.11. (i) We may assume, without loss of generality,that B(0, 4ρ) ⊂ Ω for some ρ > 0. In what follows, c(t) will denote any constantdepending only on t and Ω (or ρ) and such that c(t) is uniformly positive for tbounded and bounded away from 0. For each y ∈ R

n, let us denote by (e−tAy)t≥0

the Dirichlet heat semigroup in B(y, 3ρ).Fix y ∈ B(0, ρ) and t > 0. Since B(y, 3ρ) ⊂ Ω, the maximum principle implies

e−tAδy ≥ e−tAyδy in B(y, 3ρ), (49.20)

where δy is the Dirac measure at point y. Also, by the strong maximum principle,we have

e−tA0δ0 ≥ c(t)χB(0,2ρ). (49.21)

Since(e−tAyδy

)(x) =

(e−tA0δ0

)(x− y), it follows from (49.20) and (49.21) that

e−tAδy ≥ c(t)χB(y,2ρ) ≥ c(t)χB(0,ρ). (49.22)

On the other hand, by Hopf’s lemma (see Proposition 52.7), we have

e−tAχB(0,ρ) ≥ c(t) δ. (49.23)

Combining (49.22) and (49.23) (with t replaced by t/2), we obtain

e−tAδy = e−(t/2)A(e−(t/2)Aδy

)≥ c(t) e−(t/2)AχB(0,ρ) ≥ c(t) δ.

456 Appendices

In other words, we have shown that

G(x, y, t) = (e−tAδy

)(x) ≥ c(t) δ(x)χB(0,ρ)(y), x, y ∈ Ω, t > 0. (49.24)

Using G(x, y, t) = G(y, x, t) =(e−tAδy

)(x) =

(e−tAδx

)(y), and (49.24), (49.23)

(with t replaced by t/2), we then obtain

G(x, y, t) =(e−(t/2)A(e−(t/2)Aδx)

)(y)

≥ c(t) δ(x)(e−(t/2)AχB(0,ρ)

)(y) ≥ c(t) δ(x)δ(y),

hence assertion (i).(ii) Since K(x, y) =

∫∞0

G(x, y, t) dt, this is an immediate consequence of asser-tion (i).

49.4. Proof of Theorems 49.2, 49.3 and 49.7

We begin with the Lpδ-L

qδ-estimates. We first treat the parabolic case (Theo-

rem 49.7(i)). The elliptic case (Theorem 49.2) will next be deduced as a con-sequence.

Proof of Theorem 49.7(i). In this proof, C denotes any positive constant de-pending only on Ω (not on p, q). Let φ ∈ L2(Ω) with φ ≥ 0. Since e−tA is self-adjoint on L2(Ω) we deduce from Proposition 49.9 that, for all t > 0,

‖e−tAφ‖1 = (e−tAφ, χΩ) = (φ, e−tAχΩ) ≤ Ct−1/2 (φ, δ),

hence‖e−tAφ‖1 ≤ Ct−1/2 ‖φ‖1,δ, t > 0. (49.25)

Writing φ = φ+−φ− and using the linearity and the positivity preserving propertyof e−tA, it follows that (49.25) is true for all φ ∈ L2(Ω). Let now φ ∈ L1

δ(Ω) andtake φi ∈ L2(Ω) such that φi → φ in L1

δ(Ω). We have e−tAφi → e−tAφ in L1δ(Ω) by

(49.6), hence a.e. (up to a subsequence). By Fatou’s lemma, we infer that (49.25)is true for all φ ∈ L1

δ(Ω).Next using the L1-L∞-estimate (see Proposition 48.4), we deduce that

‖e−tAφ‖∞ = ‖e−(t/2)A(e−(t/2)Aφ)‖∞≤ (2πt)−n/2‖e−(t/2)Aφ‖1 ≤ Ct−(n+1)/2‖φ‖1,δ.

(49.26)

Now take φ ∈ Lpδ(Ω). Using (49.11) and applying (49.26) with φ replaced by |φ|p,

we get‖e−tAφ‖p

∞ = ‖|e−tAφ|p‖∞ ≤ ‖e−tA(|φ|p)‖∞≤ Ct−(n+1)/2‖|φ|p‖1,δ = Ct−(n+1)/2‖φ‖p

p,δ,

hence (49.12) for q = ∞. For p ≤ q < ∞, combining this with (49.9) yields

‖e−tAφ‖qq,δ ≤ ‖e

−tAφ‖q−p∞ ‖e−tAφ‖p

p,δ ≤ C(q−p)/pCpt−(n+1)(q−p)/2p‖φ‖qp,δ.

Raising to the power 1/q, we obtain (49.12).

Proof of Theorem 49.2. (i) Let us first assume f ∈ D(Ω). Observe that u isa solution to the inhomogeneous heat equation with initial data u and right-handside f . Therefore,

u = e−tAu +∫ t

0

e−sAf ds,

for all t > 0, by the variation-of-constants formula. Next, by (49.12) and (49.13),we have

‖e−sAf‖q,δ ≤ Cs−β‖e−(s/2)Af‖p,δ ≤ Cs−βe−λ1s/2‖f‖p,δ.

Consequently,

‖u‖q,δ ≤ Ct−βe−λ1t/2‖u‖p,δ + C(∫ t

0

s−βe−λ1s/2 ds)‖f‖p,δ,

where the integral over (0, t) is convergent due to β < 1. Estimate (49.4) forf ∈ D(Ω) follows upon letting t→∞. (Note that if q < ∞ one can also use (49.9)instead of (49.13).)

Now, in the general case f ∈ Lpδ(Ω), the conclusion follows by a density argu-

ment: Take fi ∈ D(Ω) such that fi → f in Lpδ(Ω) and let ui be the solution of

(47.8) with f replaced by fi. By (49.1) for q = 1 (which we already proved), wehave ui → u in L1(Ω), hence a.e. (up to a subsequence). Passing to the limit in‖ui‖q,δ ≤ C‖fi‖p,δ by Fatou’s lemma, the conclusion follows.

(ii) Let v ≥ 0 be the L1δ-solution of (47.8) with f replaced by f+. We have u ≤ v

by the maximum principle (cf. Theorem 49.1), hence u+ ≤ v. Estimate (49.5) thenfollows from (49.4).

Proof of Proposition 47.5. It is completely similar to that of Theorem 49.2,except that we use Propositions 48.5 and 48.4, instead of formulas (49.12) and(49.13).

Proof of (49.1) in Theorem 49.1. For any φ ∈ L1δ(Ω), using inequality (49.25),

the L1-Lq-estimate and (49.9), we get

‖e−tAφ‖q ≤ Ct−n2 (1− 1

q )‖e−(t/2)Aφ‖1 ≤ Ct−n+1

2 + n2q ‖e−(t/4)Aφ‖1,δ

≤ Ce−λ1t/4t−n+1

2 + n2q ‖φ‖1,δ.

Arguing as in the proof of Theorem 49.2(i), we then obtain (49.1).

458 Appendices

We now proceed to prove the optimality results, namely Theorem 49.3 andTheorem 49.7(ii). The proofs are based on the construction of an appropriateright-hand side of the Laplace equation (or initial data of the heat equation), withsuitable boundary singularities. It is supported in a conical subdomain of Ω withvertex at a boundary point. The following lemma provides key lower estimatesof the corresponding solutions in the same cone. This construction is used alsoin Sections 11 and 31 to show the existence of unbounded solutions of nonlinearelliptic equations and systems.

Lemma 49.13. Let n ≥ 2 and let Ω ⊂ Rn be a bounded domain of class C2+γ

for some γ ∈ (0, 1). Assume that 0 ∈ ∂Ω. Let α < n− 1. There exist R > 0 and arevolution cone Σ1 of vertex 0, with Σ := Σ1 ∩B2R ⊂ Ω, such that the function

φ := |x|−(α+2)χΣ (49.27)

belongs to L1δ(Ω) and enjoys the following properties.

(i) Denote V (t) = e−tAφ. Then

V (x, t) ≥ Ct−(α+2)/2 (49.28)

for all x, t such that x ∈ Σ, |x| ≤ R and σ|x| ≤√

t ≤ 2σ|x|, where σ > 0 is aconstant.(ii) The L1

δ-solution U > 0 of (47.8) with f = φ satisfies

U ≥ C|x|−αχΣ. (49.29)

Proof. Write x = (x1, x′), x′ = (x2, . . . , xn). Since Ω is a C2-domain, we may

assume without loss of generality that Ω contains the (truncated) revolution cone

Σ0 :=x : |x′| ≤ 2θx1, |x| ≤ 3R

,

for some θ, R > 0. Next define

Σ1 :=x : |x′| ≤ θx1, Σ := Σ1 ∩B2R,

and let φ be defined by (49.27). The fact that φ ∈ L1δ will follow from Lemma 49.14

below.Let the constant c2 ≥ 2 be given by Proposition 49.10. We observe that there

exists σ = σ(θ) ∈ (0, 1/c2) such that

δ(x) ≥ dist(x, Σc0) ≥ 2c2σ|x|, for all x ∈ Σ. (49.30)

(Indeed, dist(x, z : |z′| = 2θz1) ≥ |x| sin(β − β′), where β = arctan(2θ), β′ =arctan θ, and dist(x, z : |z| = 3R) ≥ R ≥ |x|/2.)

Let now x, t satisfy x ∈ Σ, |x| ≤ R and σ|x| ≤√

t ≤ 2σ|x|. In particular, wehave t ≤ (2σ|x|)2 ≤ R2 and, by (49.30), δ(x) ≥ c2

√t. By Proposition 49.10, it

follows that

V (x, t) =∫

Ω

G(x, y, t)φ(y) dy ≥ c1t−n/2

∫|x−y|≤√

t

|y|−(α+2)χΣ(y) dy.

Observe that, due to x ∈ Σ, |x| ≤ R and t ≤ R2, we have Σ ∩ B(x,√

t) ⊃(x + Σ) ∩ B(x,

√t), hence meas(Σ ∩ B(x,

√t)) ≥ Ctn/2. Since σ|x| ≤

√t ≤ 2σ|x|

(with 0 < σ < 1/2) and |x− y| ≤√

t imply c√

t ≤ |y| ≤ C√

t, we obtain

V (x, t) ≥ Ct−n/2 t−(α+2)/2 meas(Σ ∩B(x,√

t)) ≥ Ct−(α+2)/2. (49.31)

This proves (i).Let x ∈ Σ. If |x| ≤ R, by (49.28), we have

U(x) =∫ ∞

0

V (x, t) dt ≥∫ 4σ2|x|2

σ2|x|2Ct−(α+2)/2 dt ≥ C|x|−α.

If |x| ≥ R, then δ(x) ≥ 2c2σR due to (49.30). By Remark 49.12(i), it follows thatU(x) ≥ C with C > 0 independent of x. Thus (ii) is proved.

As for the integrability properties of the functions φ, U , V , we have the followingsimple lemma.

Lemma 49.14. Let Ω, α, φ, U be as in Lemma 49.13.(i) Assume α > −2. The function φ ∈ Lp

δ(Ω) if and only if p < (n + 1)/(α + 2).(ii) Assume α > 0. If q ≥ (n + 1)/α, then U ∈ Lq

δ(Ω).

(iii) For 1 ≤ q ≤ ∞, there holds ‖V (t)‖q,δ ≥ Ctn+12q −α+2

2 for t > 0 small.

Proof. (i) We have ‖φ‖pp,δ = C

∫Σ |x|−(α+2)pδ(x) dx. By (49.30) and δ(x) ≤ |x|,

the last integral has the same nature (finite or infinite) as∫Σ

|x|1−(α+2)p dx =∫ 2R

0

rn−(α+2)p dr

∫Σ′

dω,

where Σ′ = x/|x| ∈ Sn−1 : x ∈ Σ \ 0. Therefore, φ ∈ Lpδ if and only if

p < (n + 1)/(α + 2).(ii) In view of (49.29), this follows from assertion (i).

(iii) Due to (49.28) we may assume q < ∞. Let A(t) = x ∈ Σ : σ|x| ≤√

t ≤2σ|x|. For t < σ2R2, we have A(t) ⊂ BR. By (49.28) and (49.30), it follows that∫

Ω

V q(x, t)δ(x) dx ≥ Ct−α+2

2 q

∫A(t)

δ(x) dx ≥ Ct12−α+2

2 q

∫A(t)

dx

≥ Ct12−α+2

2 q

∫ √t/σ

√t/2σ

rn−1 dr

∫Σ′

dω = Ctn+1

2 −α+22 q.

460 Appendices

After these preparations, we can now easily conclude.

Proof of Theorem 49.3. By assumption, one can choose α ∈ (0, n − 1) suchthat n+1

q < α < n+1p − 2. The result then follows from Lemmas 49.13 and 49.14(i)

and (ii).

Proof of Theorem 49.7(ii). Choose α > −2 such that n+1p − 2 − 2ε < α <

n+1p −2. Then n+1

2q −α+2

2 < −β + ε and the result follows from Lemmas 49.13 and49.14(i) and (iii).

49.5. The heat equation in uniformly local Lebesgue spaces

We have the following smoothing property for the linear heat equation in uniformlylocal spaces.

Proposition 49.15. Let 1 ≤ p < ∞.(i) The heat semigroup on R

n, given by e−tAφ = Gt ∗φ, is well defined on Lpul and

e−tA(Lpul) ⊂ L∞ for all t > 0.

(ii) Let 0 < T < ∞, p ≤ q ≤ ∞ and φ ∈ Lpul. Then

‖e−tAφ‖q,ul ≤ C(n, p, q, T )t−n2 ( 1

p− 1q )‖φ‖p,ul, 0 < t ≤ T.

We use the following simple lemma.

Lemma 49.16. Let 1 ≤ p <∞. The norms ‖ · ‖p,ul and ‖ · ‖p,∗, where

‖φ‖p,∗ := supa∈Rn

(∫Rn

|φ(a− y)|pG1(y) dy

)1/p

= ‖G1 ∗ |φ|p‖1/p∞ ,

are equivalent on Lpul.

Proof. On the one hand, we have∫Rn

|φ(y)|pG1(a− y) dy ≥ (4π)−n/2e−1/4

∫|y−a|<1

|φ(y)|p dy,

hence ‖φ‖p,∗ ≥ c‖φ‖p,ul. On the other hand, there holds∫Rn

|φ(y)|pG1(y) dy ≤∑

k∈Zn

exp[− 1

4

( 2|k|√n− 1)2+

] ∫|y−2k/

√n|<1

|φ(y)|p dy

≤ C supa∈Rn

(∫|y−a|<1

|φ(y)|p dy)

with C =∑

k∈Zn exp[− 1

4

( 2|k|√n− 1

)2+

]< ∞. After a translation, this implies

‖φ‖p,∗ ≤ C1/p‖φ‖p,ul.

Proof of Proposition 49.15. Let 1 ≤ p < ∞ and φ ∈ Lpul. We may assume

φ ≥ 0 without loss of generality. Moreover, by the semigroup property (48.6), it issufficient to consider T = 1 and 0 < t ≤ 1.

By Jensen’s inequality and ‖Gt‖L1 = 1, it follows that

(Gt ∗ φ)p ≤ Gt ∗ φp. (49.32)

On the other hand, we have

Gt ≤ t−n/2G1, 0 < t ≤ 1. (49.33)

Using (49.32) and (49.33), Lemma 49.16 implies in particular that e−tAφ is welldefined as an element of L∞, hence assertion (i).

From (49.32) and ‖Gt‖L1 = 1, we deduce

‖G1 ∗ (e−tAφ)p‖∞ = ‖G1 ∗ (Gt ∗ φ)p‖∞ ≤ ‖G1 ∗Gt ∗ φp‖∞= ‖Gt ∗G1 ∗ φp‖∞ ≤ ‖G1 ∗ φp‖∞

hence

‖e−tAφ‖p,∗ ≤ ‖φ‖p,∗, t > 0. (49.34)

On the other hand, (49.32) and (49.33) imply

‖e−tAφ‖∞ ≤ ‖Gt ∗ φp‖1/p∞ ≤ t−n/2p‖G1 ∗ φp‖1/p

∞ = t−n/2p‖φ‖p,∗. (49.35)

Now for p ≤ q < ∞, it follows from (49.34) and (49.35) that

‖e−tAφ‖qq,∗ = ‖G1 ∗ (e−tAφ)q‖∞ ≤ ‖G1 ∗ (e−tAφ)p‖∞‖e−tAφ‖q−p

∞= ‖e−tAφ‖p

p,∗‖e−tAφ‖q−p∞ ≤ t−(n/2)(q/p−1)‖φ‖q

p,∗.

This along with (49.35) and Lemma 49.16 yields assertion (ii).

462 Appendices

50. Appendix D: Poincare, Hardy-Sobolev, andother useful inequalities

50.1. Basic inequalities

In this subsection we recall some basic inequalities which we frequently use.

Young’s inequality. Let 1 0 and let q = p′ = p/(p− 1). Then

xy ≤ εpxp

p+

ε−qyq

q, x, y > 0.

In what follows, Ω is an arbitrary domain in Rn.

Holder’s inequality. Let 1 ≤ p ≤ ∞ and q = p′ = p/(p− 1). Then

‖uv‖1 ≤ ‖u‖p‖v‖q, u ∈ Lp(Ω), v ∈ Lq(Ω).

A useful consequence is the following interpolation inequality. Let 1 ≤ p < r <q ≤ ∞. If u ∈ Lp ∩ Lq(Ω), then u ∈ Lr(Ω) and

‖u‖r ≤ ‖u‖θp‖u‖1−θ

q , where θ =(1

r− 1

q

)(1p− 1

q

)−1

∈ (0, 1).

Jensen’s inequality. Assume that F : R → [0,∞) is a convex function, and thatw : Ω → [0,∞] is measurable and satisfies

∫Ω

w(x) dx = 1. If u is a measurablefunction on Ω such that uw, F (u)w ∈ L1(Ω), then

F(∫

Ω

u(x)w(x) dx)≤∫

Ω

F (u(x))w(x) dx.

Sobolev’s inequality. Let 1 ≤ p < n and denote p∗ = np/(n− p). Then

‖u‖p∗ ≤ C(n, p)‖∇u‖p, u ∈ W 1,p0 (Ω). (50.1)

50. Appendix D: Poincare, Hardy-Sobolev, and other useful inequalities 463

50.2. The Poincare inequality

Let Ω be an arbitrary domain in Rn and let 1 ≤ q < ∞. The Poincare inequality

in W 1,q0 (Ω) is the statement that

‖v‖q ≤ Cq(Ω)‖∇v‖q, for all v ∈ W 1,q0 (Ω). (50.2)

It is well known (see e.g. [90]) that (50.2) holds in any bounded domain, or moregenerally in any domain which is bounded in one direction. However, since this isa basic inequality in the study of elliptic and parabolic problems, it is importantto have a characterization of those domains Ω such that (50.2) is true. It turns outthat there is a simple geometric necessary condition, which is also almost sufficient.Moreover the equivalence is true for uniformly regular domains.

To this end, let us introduce the notion of inradius ρ(Ω) of a domain Ω:

ρ(Ω) = supr > 0 : Ω contains a ball of radius r

= sup

x∈Ωdist(x, ∂Ω).

We also define the strict inradius ρ′(Ω) ≥ ρ(Ω), given by:

ρ′(Ω) = infR > 0 : ∃ε > 0 such that for any ball B of radius R,

B ∩ Ωc contains a ball of radius ε.

The relation between Poincare inequalities and the inradius and strict inradiusis given by the following result.

Proposition 50.1. Let Ω be an arbitrary domain in Rn.

(i) If (50.2) holds for some q ∈ [1,∞), then ρ(Ω) < ∞.(ii) If ρ′(Ω) <∞, then (50.2) holds for all 1 ≤ q <∞.(iii) Assume that Ω is uniformly regular or, more generally, that Ω satisfies auniform exterior cone condition. Then for all 1 ≤ q < ∞, (50.2) holds if and onlyif ρ(Ω) < ∞.

Examples 50.2. Let us give some simple examples concerning inradius and strictinradius in the case of unbounded domains.

(a) If Ω is contained in a strip, then ρ and ρ′ are both finite, while if Ω containsan infinite cone, then they are both infinite.

(b) If Ω is the complement of a periodic net of points, Ω = Rn \

⋃z∈Zn

Rz, for

some R > 0, then ρ(Ω) = n1/2R/2, ρ′(Ω) =∞.

(c) If Ω is the complement of a periodic net of balls of constant radius, Ω =R

n \⋃

z∈Zn

B(Rz, ε), for some 0 < ε < R/2, then ρ(Ω) = ρ′(Ω) = n1/2R/2− ε.

Part (i) of Proposition 50.1 is easy. The idea of the proof of part (ii) is due to[4, Lemma 7.4 p. 75], where this is done for q = 2 (see [481] for the general case).

464 Appendices

On the other hand, Proposition 50.1 (for all q) can be proved as a consequence ofmore general and difficult results, where the domain need not be uniformly regular(see [336, Corollary 2]). See also [154, Section 1.5] and the references therein forrelated results.

Proof of Proposition 50.1. (i) Assume ρ(Ω) =∞. This means that Ω containssome ball Bj = B(xj , j) for all j ≥ 1. Fixing a test-function w ∈ D(Rn), w ≥ 0,w ≡ 0 with supp(w) ⊂ B(0, 1), and setting wj(x) = w((x − xj)/j), we get thatwj ∈ D(Ω) and that

‖wj‖q = jn/q‖w‖q and ‖∇wj‖q = j(n/q)−1‖∇w‖q.

Consequently, (50.2) is false.(ii) By density, it obviously suffices to consider the case v ∈ D(Ω).

Applying the definition of ρ′ := ρ′(Ω), we may choose ε ∈ (0, 1) such that forany ball B of radius ρ′ + 1, B ∩Ωc contains a ball of radius ε. Let Q be any cubeof edge 2(ρ′ + 1), such that Q ∩ Ω = ∅. By translation we may assume that Qis centered at the origin. By the definition of ρ′, there exists a point a such thatB(a, ε) ⊂ B(0, ρ′ + 1) ∩ Ωc. In particular, B(a, ε) ∩ Ω = ∅ and d(0, a) < ρ′ + 1(hence a ∈ Q).

Using polar coordinates about a, denoted by (r, ω), we may represent the cubeQ by the set Q = (r, ω) : ω ∈ Sn−1, 0 ≤ r < R(ω), where R(ω) is some(continuous nonnegative) function. Using supp(v) ⊂ Ω and B(a, ε)∩Ω = ∅, we get

∫Q

|v(x)|q dx =∫

Sn−1

∫ R(ω)

ε

|v(r, ω)|qrn−1 dr dω.

Now, for all x ∈ Q, there holds

d(a, x) ≤ d(a, 0) + d(0, x) ≤ R := (1 + n1/2)(ρ′ + 1),

hence R(ω) ≤ R. Using Holder’s inequality, we have, for all (r, ω) ∈ Q,

|v(r, ω)|q =∣∣∣∫ r

ε

vr(σ, ω) dσ∣∣∣q ≤ Rq−1

∫ R(ω)

ε

|vr(σ, ω)|q dσ

≤ ε1−nRq−1

∫ R(ω)

ε

|vr(σ, ω)|qσn−1 dσ.

It follows that∫Sn−1

∫ R(ω)

ε

|v(r, ω)|qrn−1 dr dω ≤ Rn+q−1

nεn−1

∫Sn−1

∫ R(ω)

ε

|vr(r, ω)|qrn−1 dr dω,

50. Appendix D: Poincare, Hardy-Sobolev, and other useful inequalities 465

hence ∫Q

|v(x)|q dx ≤ Rn+q−1

nεn−1

∫Q

|∇v(x)|q dx.

Dividing Rn into a periodic net of cubes of edge 2(ρ′ + 1), and summing this

inequality over all cubes yields the same inequality with Rn instead of Q, that is

(50.2), with

Cq(Ω) = (1 + n1/2)1+(n−1)/qn−1/q(ρ′(Ω) + 1)(

2 + ρ′(Ω)ε

)(n−1)/q

.

(iii) This follows immediately from (i) and (ii).

50.3. Hardy and Hardy-Sobolev inequalities

The following lemma is a simple version of the Hardy inequality.

Lemma 50.3. Let Ω ⊂ Rn be a bounded domain of class C1. Then there exists a

positive constant C = C(Ω) such that ‖u/δ‖2 ≤ C‖∇u‖2 for all u ∈ W 1,20 (Ω).

Proof. First consider the case n = 1, Ω = (0, 1) and assume u(x) = 0 for x ∈(0, ε]. Then integration by parts and the Cauchy inequality imply∫ 1

0

u2

x2dx = − 1

xu2(x)

∣∣∣1ε

+ 2∫ 1

0

1x

uu′ dx ≤ 2(∫ 1

0

u2

x2dx)1/2(∫ 1

0

(u′)2 dx)1/2

,

hence ∥∥∥u

x

∥∥∥2≤ 2‖u′‖2. (50.3)

If, in general, u ∈ W 1,20 (0, 1), then there exist uk ∈ D(0, 1) such that uk → u a.e.

and in W 1,2(0, 1). Fatou’s lemma and (50.3) imply∫ 1

0

u2

x2dx ≤ lim inf

k→∞

∫ 1

0

u2k

x2dx ≤ lim inf

k→∞4‖u′

k‖22 = 4‖u′‖22.

This inequality and the symmetric estimate ‖u/(1 − x)‖2 ≤ 2‖u′‖2 imply theassertion in the case Ω = (0, 1).

Let n > 1, Ω = (0, 1)n and u ∈ D(Ω). Writing x = (x1, x′), x′ = (x2, x3, . . . , xn),

integrating the inequality∫ 1

0

u2(x1, x′)

x21

dx1 ≤ 4∫ 1

0

( ∂u

∂x1(x1, x

′))2

dx1

466 Appendices

over x′ ∈ (0, 1)n−1 and using Fubini’s theorem we obtain∫Ω

u2

x21

dx ≤ 4∫

Ω

( ∂u

∂x1

)2

dx ≤ 4∫

Ω

|∇u|2 dx.

Similarly as above, this implies the assertion in the case Ω = (0, 1)n.If Ω ⊂ R

n is a C1 bounded domain, then one can use standard localizationarguments (partition of unity and flattening the boundary ∂Ω) in order to provethe assertion.

A combination of Lemma 50.3 and the Sobolev inequality (50.1) with p = 2yields the following Hardy-Sobolev inequality.

Lemma 50.4. Let Ω ⊂ Rn be a bounded domain of class C1, n ≥ 3, τ ∈ [0, 1],

with 1/q = 1/2∗ + τ/n. Then there exists a positive constant C = C(Ω, τ) suchthat ‖u/δτ‖q ≤ C‖∇u‖2 for all u ∈W 1,2

0 (Ω).

Proof. Due to Lemma 50.3 and the Sobolev inequality we may assume τ ∈ (0, 1).Setting m := 2/τ and s := 2∗/(1 − τ) we have 1/q = 1/m + 1/s and Holder’sinequality implies∥∥∥ u

δτ

∥∥∥q≤∥∥∥uτ

δτ

∥∥∥m‖u1−τ‖s =

∥∥∥u

δ

∥∥∥τ

mτ‖u‖1−τ

s(1−τ) ≤ C‖∇u‖τ2‖∇u‖1−τ

2 = C‖∇u‖2,

where we used Lemma 50.3 and the Sobolev inequality again.

Remark 50.5. One can easily see that if n = 2 or n = 1, then the assertion ofLemma 50.4 remains true for any q ≥ 1 satisfying 1/q > τ/2 or 1/q > τ − 1/2,respectively.

51. Appendix E: Local existence, regularity andstability for semilinear parabolic problems

51.1. Analytic semigroups and interpolation spaces

In this subsection we recall some basic facts on strongly continuous analytic semi-groups and interpolation spaces. We refer to [77], [273], [410], [13], [14], [16], [344]and [513] for details.

Let X0 be a Banach space endowed with the norm | · |0 and let A be a closedlinear densely-defined operator in X0. Denote by D(A) the domain of definitionof A endowed with the graph norm ‖x‖A := |x|0 + |Ax|0 and let X1 be a Banachspace endowed with the norm | · |1 and satisfying X1

.= D(A). Then −A generates

51. Appendix E: Local existence, regularity and stability 467

a C0 analytic semigroup e−tA in X0 if and only if there exist C > 0 and ω ∈ R

such that ω + A : X1 → X0 is an isomorphism and

|λ||x|0 + |x|1 ≤ C|(λ + A)x|0 for all x ∈ X1, Re λ ≥ ω. (51.1)

Setω(−A) := supRe λ : λ ∈ σ(−A),

where σ(−A) denotes the spectrum of −A. If −A generates a C0 analytic semi-group in X0 and ω > ω(−A), then there exists C > 0 such that (51.1) is true.

Unless explicitly stated otherwise, throughout the rest of Appendix E we shallassume that

X0 is a reflexive Banach space,

−A generates a C0 analytic semigroup in X0,

ω > ω(−A).

⎫⎪⎬⎪⎭ (51.2)

We will also consider the scale of spaces Xα and operators Aα, α ∈ [−1, 1], definedas follows.

Let X−1 be the completion of X0 endowed with the norm |x|−1 := |(ω+A)−1x|0.Given θ ∈ (0, 1), set Xθ := (X0, X1)θ and X−1+θ := (X−1, X0)θ, where (·, ·)θ iseither the complex interpolation functor [·, ·]θ or any of the real interpolationfunctors (·, ·)θ,p, 1 < p < ∞. Given θ ∈ [0, 1], let Aθ be the Xθ-realization ofA (i.e. Aθx = Ax for x ∈ D(Aθ) := x ∈ Xθ : Ax ∈ Xθ) and let A−1+θ

be the closure of A in X−1+θ (A is closable in X−1+θ). The following theoremis a consequence of [14, Theorems 8.1, 8.3 and Corollary 8.2], [16, TheoremsII.1.2.2, III.2.5.6, III.3.4.1, III.4.10.7 and Chapter V] and the proof of [344, Propo-sition 4.2.1].

Theorem 51.1. Let −1 ≤ β ≤ α ≤ 1. Then the following assertions are true.(i) The space Xα is densely embedded in Xβ; the embedding Xα → Xβ is compactprovided A has compact resolvent and α > β.(ii) We have

(Xβ , Xα)η+ → X(1−η)β+ηα → (Xβ , Xα)η−

for any 0 < η− < η < η+ < 1 and the embeddings are dense (almost reiterationproperty).(iii) Aα is the Xα-realization of Aβ and σ(Aα) = σ(Aβ).(iv) −Aα generates a C0 analytic semigroup e−tAα in Xα. In addition,

e−tAα = e−tAβ |Xα ,

and there exists C = C(ω, A) > 0 such that

‖e−tAβ‖L(Xβ ,Xα) ≤ Ctβ−αeωt for all t > 0. (51.3)

468 Appendices

(v) Let u0 ∈ X0, η, ε > 0, η + ε < 1, f ∈ Cε([0, T ], Xη) + C([0, T ], Xη+ε). Thenthere exists a unique u ∈ C([0, T ], X0)∩C1((0, T ], X0)∩C((0, T ], X1) which solvesthe linear Cauchy problem

u + Au = f in (0, T ], u(0) = u0. (51.4)

In addition, u satisfies the variation-of-constants formula

u(t) = e−tAu0 +∫ t

0

e−(t−s)Af(s) ds.

If u0 ∈ Xη, then u ∈ C([0, T ], Xη). If u0 ∈ X1, then u ∈ C1([0, T ], X0). If ρ ∈(0, 1), θ ∈ [0, 1] and f ∈ Cρ((0, T ], Xθ), then u ∈ C1+ρ((0, T ], Xθ).(vi) Let X0 be a UMD space, ω(−A) < 0,

‖Ait‖ ≤Meθ|t| for some M > 0, θ ∈ [0, π/2) and all t ∈ R, (51.5)

u0 ∈ X0 := (X0, X1)1−1/p,p and f ∈ Xf := Lp([0, T ], X0), where 1 < p < ∞. Thenthe Cauchy problem (51.4) possesses a unique solution u ∈ X := Lp([0, T ], X1) ∩W 1,p([0, T ], X0) and

‖u‖X ≤ C(‖u0‖X0 + ‖f‖Xf),

where C does not depend on u0, f and T .(vii) Let α ≥ 0, α− 1 < γ < α, f ∈ L∞((0, T ), Xα−1), and

v(t) :=∫ t

0

e−(t−s)Af(s) ds.

Then v ∈ Cα−γ([0, T ], Xγ).

The definition and properties of UMD spaces can be found in [16, SectionsIII.4.4–III.4.5]. For example, the Lebesgue spaces Lq(Ω), 1 < q < ∞, and Hilbertspaces are UMD spaces. For sufficient conditions for the boundedness of imaginarypowers of A see Remark 51.5 and also [18], [161], [166], [169], [430], [476].

If α ∈ [−1, 1] and no confusion seems likely, then we will shortly write A ande−tA instead of Aα and e−tAα , respectively. We will also denote by | · |α the normin Xα.

Remarks 51.2. (i) In [273] the author uses the fractional power spaces Xα,α ≥ 0, instead of the interpolation spaces Xα. However, if the operator A hasbounded imaginary powers (that is if the estimate in (51.5) is true for some M > 0,θ ≥ 0 and all t ∈ R), then the fractional power spaces are equivalent to theinterpolation spaces obtained by using the complex interpolation functor [·, ·]θ, see[16, Theorem V.1.5.4]. In the general case, we still have Xα → Xβ and Xα → Xβ

whenever 1 ≥ α > β ≥ 0.

(ii) The advantage of interpolation and extrapolation spaces becomes evidentin Subsection 51.5 where we deal with singular initial data. Extrapolation spacesalso naturally appear if one uses semigroup approach to problems with nonlinearboundary conditions (see [14], [15], [17], [444]).

We will also need the following interpolation estimate (see [440, Proposition 2.1]and the references therein for a more general statement). We say that (E0, E1) isan interpolation couple of Banach spaces if E0, E1 are Banach spaces and thereexists a locally convex space E such that E0, E1 → E.

Proposition 51.3. Let (E0, E1) be an interpolation couple of Banach spaces. Let1 ≤ p0, p1 < ∞, θ ∈ (0, 1), 1/pθ = (1−θ)/p0+θ/p1, s := 1−θ, Eθ := (E0, E1)θ,pθ

.Then

W 1,p0([0, T ], E0) ∩ Lp1([0, T ], E1) →W s,pθ ([0, T ], Eθ)

and the norm of this embedding can be estimated by a constant C(T0) for allT ∈ (0, T0].

If E1 is compactly embedded in E0 and s < 1 − θ, then the above embedding iscompact.

If p > 1, r ≥ 1 and Ω ⊂ Rn is open, then Proposition 51.3 implies

W 1,2([0, T ], L2(Ω)) ∩ L(p+1)r([0, T ], Lp+1(Ω)) → L∞([0, T ], Lq(Ω)) (51.6)

for any q ∈ [2, p+1− (p−1)/(r+1)) (see [440] for details and see [114] for a directproof).

Examples 51.4. (See [13] and [14].)(i) Let Ω ⊂ R

n be uniformly regular of class C2, 1 < q < ∞, X0 = Lq(Ω), X1 =W 2,q ∩W 1,q

0 (Ω) (this choice of X1 corresponds to Dirichlet boundary conditions).Let A be the unbounded linear operator in X0 with domain of definition X1 definedby

Au = −n∑

i,j=1

aij∂2

∂xi∂xju +

n∑i=1

bi∂

∂xiu + cu,

where aij , bi, c ∈ BUC(Ω) and aij = aji are uniformly elliptic. Then −A generatesa C0 analytic semigroup in X0. Let (·, ·)θ be the complex interpolation functor ifθ = 1/2 and the real interpolation functor (·, ·)θ,q otherwise. Then

Xθ = Xθ(q).= u ∈W 2θ,q(Ω) : u = 0 on ∂Ω if 2θ > 1/q,

W 2θ,q(Ω) if 1/q > 2θ ≥ 0,

X1/2q(q) →W 1/q,q(Ω), and

Xθ(q).=(X−θ(q′)

)′ if θ < 0. (51.7)

470 Appendices

(ii) If we set X1 = u ∈ W 2,q(Ω) : ∂u/∂n = 0 on ∂Ω (Neumann boundaryconditions), then the assertions in (i) remain true with

Xθ = Xθ(q).= u ∈W 2θ,q(Ω) : ∂u/∂n = 0 on ∂Ω if 2θ > 1 + 1/q,

W 2θ,q(Ω) if 1 + 1/q > 2θ ≥ 0,

X1/2+1/2q(q) →W 1+1/q,q(Ω), and (51.7).

Remark 51.5. Assume that Ω, A and Xα, α ∈ [−1, 1], are as in Examples 51.4.Then A satisfies (51.5) (see [17] and cf. also [161] and the references therein). If usolves (51.4), 1 0 does not depend on f, u0 and T .

In what follows we will also need the following singular Gronwall inequality (see[16, Theorem 3.3.1]).

Proposition 51.6. Let α, β ∈ [0, 1) and ε > 0. Then there exists a positiveconstant c := c(α, β, ε) such that the following is true:

If A, B > 0 and u : [0, T )→ R+ satisfies [t → tβu(t)] ∈ L∞loc([0, T )) and

u(t) ≤ At−β + B

∫ t

0

(t− τ)−αu(τ) dτ, for a.a. t ∈ (0, T ),

thenu(t) ≤ At−β

(1 + cBt1−αe(1+ε)µt

)for a.a. t ∈ (0, T ),

where µ := (Γ(1 − α)B)1/(1−α).

51.2. Local existence and regularity for regular data

Recall that we assume (51.2) and that Xα, α ∈ [−1, 1] denote the correspondinginterpolation-extrapolation scale of spaces. The proofs of the following theoremand Theorems 51.17, 51.19, 51.21, 51.25, 51.33 below are based on well-known andfrequently used ideas (see [273], [344], for example).

Theorem 51.7. Fix 1 ≥ α > β ≥ 0 and assume that F : Xβ → Xα−1 is locallyLipschitz continuous, uniformly on bounded subsets of Xβ. Let u0 ∈ Xβ. Thenthere exists T = T (|u0|β) > 0 such that the integral equation

u(t) = e−tAu0 +∫ t

0

e−(t−s)AF (u(s)) ds (51.9)


has a unique solution u ∈ C([0, T ], Xβ). In addition, there exists C = C(A) > 0such that |u(t)|β ≤ C|u0|β + 1 for all t ∈ [0, T ].

If γ ∈ [β, α), γ > α− 1, then u ∈ Cα−γ((0, T ], Xγ). Moreover we have the fol-lowing continuous dependence property: if γ ∈ [β, α) and u and u are two solutionswith initial data u0 and u0, respectively, then there exist T = T (|u0|β , |u0|β) > 0and C > 0 independent of the initial data such that

|u(t)− u(t)|γ ≤ Ctβ−γ |u0 − u0|β for all t ∈ (0, T ]. (51.10)

Finally, the solution can be continued on the maximal existence interval [0, Tmax),where either Tmax = ∞ or limt→Tmax |u(t)|β = ∞.

Proof. Due to (51.3), there exists CA > 0 such that

‖e−tA‖L(Xα1 ,Xα2 ) ≤ CAtα1−α2 for all t ∈ (0, 1] and − 1 ≤ α1 ≤ α2 ≤ 1.

(51.11)Let M > 2CA|u0|β . The assumptions on F guarantee the existence of CF =CF (M) > 0 such that

|F (u)|α−1 ≤ CF and |F (u)− F (v)|α−1 ≤ CF |u− v|β (51.12)

for all u, v ∈ Xβ satisfying |u|β , |v|β ≤ M . Assume T ∈ (0, 1] and let BM = BM,T

denote the closed ball in the Banach space YT := C([0, T ], Xβ) with center 0 andradius M . We will use the Banach fixed point theorem for the mapping Φu0 :BM → BM , where

Φu0(u)(t) := e−tAu0 +∫ t

0

e−(t−s)AF (u(s)) ds. (51.13)

Let u ∈ BM . Then

|Φu0(u)(t)|β ≤ ‖e−tA‖L(Xβ ,Xβ)|u0|β +∫ t

0

‖e−(t−s)A‖L(Xα−1,Xβ)|F (u(s))|α−1 ds

≤ CA|u0|β + CACF

∫ t

0

(t− s)α−1−β ds

≤ 12M +

CACF

α− βT α−β ≤M,

provided T ≤ τ0, where τ0 = τ0(M) > 0 is small enough. Hence Φu0 maps BM

into BM for T ≤ τ0. Given u, v ∈ BM , we have

|(Φu0(u)− Φu0(v)

)(t)|β ≤

∫ t

0

‖e−(t−s)A‖L(Xα−1,Xβ)|F (u(s))− F (v(s))|α−1 ds

≤ CACF

∫ t

0

(t− s)α−1−β |u(s)− v(s)|β ds

≤ CACFT α−β

α− β‖u− v‖YT ≤

12‖u− v‖YT ,

472 Appendices

provided T ≤ τ1, where τ1 = τ1(M) > 0 is small enough. Consequently, Φu0 is acontraction in BM,T for T ≤ τ2 := min(τ0, τ1) and possesses a unique fixed pointu in BM,T . It is easily seen that this solution of (51.9) is unique in YT . Notice alsothat τ2 = τ2(|u0|β) if we fix M = 2CA|u0|β + 1, for example.

Let γ ∈ [β, α). If γ > α−1, then u ∈ Cα−γ((0, T ], Xγ) due to Theorem 51.1(vii).Next assume u0, u0 ∈ Xβ and fix M > 2CA max(|u0|β , |u0|β). Set

u0(t) := e−tAu0, u0(t) := e−tAu0,

uk+1 := Φu0(uk), uk+1 := Φu0(u

k), k = 0, 1, 2, . . . .

Due to the above existence proof, uk and uk converge to the solutions u and u inBM for T small enough. Now (51.11) implies the following inequality for k = 0and all t ∈ [0, T ]:

|uk(t)− uk(t)|β ≤ 2CA|u0 − u0|β . (51.14)

Assume that (51.14) is true for some k ≥ 0. Then

|uk+1(t)− uk+1(t)|β ≤ ‖e−tA‖L(Xβ ,Xβ)|u0 − u0|β

+∫ t

0

‖e−(t−s)A‖L(Xα−1,Xβ)|F (uk(s)) − F (uk(s))|α−1 ds

≤ CA|u0 − u0|β + CACF

∫ t

0

(t− s)α−1−β |uk(s)− uk(s)|β ds

≤(CA + 2C2

ACFT α−β

α− β

)|u0 − u0|β ≤ 2CA|u0 − u0|β ,

provided T is small enough. Consequently, (51.14) is true for all k. Passing to thelimit we obtain

|u(t)− u(t)|β ≤ 2CA|u0 − u0|β .

Using this estimate we finally obtain

|u(t)− u(t)|γ = |Φu0(u)(t)− Φu0(u)(t)|γ≤ ‖e−tA‖L(Xβ ,Xγ )|u0 − u0|β

+∫ t

0

‖e−(t−s)A‖L(Xα−1,Xγ)|F (u(s))− F (u(s))|α−1 ds

≤ CAtβ−γ |u0 − u0|β + CACF

∫ t

0

(t− s)α−1−γ |u(s)− u(s)|β ds

≤(CA + 2C2

ACFT α−β

α− γ

)tβ−γ |u0 − u0|β ≤ 2CAtβ−γ |u0 − u0|β ,

provided T is small enough.The existence of a maximal solution follows in the same way as in the proof of

Proposition 16.1.

Remarks 51.8. (i) The solution u in Theorem 51.7 satisfies u ∈ C1((0, Tmax),Xγ−1) and

u + Aγ−1u = F (u), t ∈ (0, Tmax),

for all γ ∈ [β, α). In fact, set X0 := Xγ−1, X1 := Xγ and let Xη, η ∈ [−1, 1],be the corresponding interpolation-extrapolation scale. Then Xα−1 → Xη for anyη ∈ (0, α − γ) due to Theorem 51.1(ii), hence F (u) ∈ C([0, Tmax), Xη). Now theassertion follows from Theorem 51.1(v).

(ii) It is straightforward to check that all statements in Theorem 51.7 re-main true for nonautonomous nonlinearities of the form F = F (t, u) providedF : [0,∞) × Xβ → Xα−1 is measurable in t, locally Lipschitz continuous in u(uniformly on bounded subsets of [0,∞)×Xβ) and F (·, 0) is bounded in Xα−1 onbounded subsets of [0,∞).

Similarly, if we assume that D ⊂ Xβ is open, F : D → Xα−1 is locally Lipschitzcontinuous (uniformly on bounded sets M ⊂ D satisfying distXβ

(M, ∂D) > 0) andu0 ∈ D, then there exists a unique maximal solution u ∈ C([0, Tmax), D) and (atleast) one of the following possibilities occurs: (a) Tmax =∞; (b) limt→Tmax |u(t)|β= ∞; (c) lim inft→Tmax distXβ

(u(t), ∂D) = 0.Finally, if ∞ > r > 1/(α − β), F : C([0, T ], Xβ) → Lr([0, T ], Xα−1) is uni-

formly Lipschitz continuous on bounded sets and has the Volterra property (thatis F (u)|[0,t] depends on u|[0,t] only), and u0 ∈ Xα−1/r, then the problem

ut + Au = F (u), t > 0,

u(0) = u0,(51.15)

has a unique maximal strong solution u ∈ C([0, Tmax), Xβ)∩W 1,rloc ([0, Tmax), Xβ−1)

due to [21, Theorem 2.3]. Strong solution means that the equation ut +Au = F (u)is satisfied for a.e. t. Notice also that F (u) ∈ Lr

loc([0, Tmax), Xα−1) is well definedfor u ∈ C([0, Tmax), Xβ) due to the Volterra property of F . Additional regularityand stability results for solutions of (51.15) can be found in [21]. In particular,u ∈ Cρ([0, Tmax), Xβ) ∩ W 1,r

loc ([0, Tmax), Xγ−1) for all ρ < α − β − 1/r and γ ∈(β, α) and the solution u is global (Tmax = T and u ∈ C([0, T ], Xβ)) wheneverF (u) ∈ Lr([0, Tmax), Xα−1).

(iii) Let α, β, γ, F and u0 be as in Theorem 51.7, and Tmax = Tmax(u0) be themaximal existence time of the solution u of (51.9). Fix t ∈ (0, Tmax). Using (51.10)one can easily prove the existence of positive constants C, ε (depending on t andmax0≤s≤t |u(s)|β) such that Tmax(u0) > t and

|u(t)− u(t)|γ < C|u0 − u0|β

for any u0 ∈ Xβ satisfying |u0 − u0|β < ε.(iv) Let X0 be a (reflexive) ordered Banach space with a total positive cone P0

and let the semigroup e−tA0 be positive (note that P0 is total if P0−P0 is dense in

474 Appendices

X0). Define positive cones Pθ in Xθ, θ ∈ [−1, 1] as follows: Pθ = P ∩Xθ if θ > 0, Pθ

is the closure of P in Xθ if θ < 0. Then Xθ become ordered Banach spaces and thesemigroups e−tAθ are positive. If, in addition, F maps Pβ into Pα−1 and u0 ∈ Pβ ,then the corresponding solution u is obviously nonnegative. In fact, u = lim uk,where u0 = e−tAu0 ≥ 0 and uk+1 = Φu0u

k ≥ 0 whenever uk ≥ 0.In particular, if e−tA is positive, u0 ∈ Pβ and F : Pβ → Pα−1, then F need not

be defined for u /∈ Pβ (any regular extension of F to Xβ leads to the same positivesolution u).

(v) A simple modification of the proof of Theorem 51.7 shows that the assump-tion β ≥ 0 can be replaced with β ≥ −1.

Example 51.9. Let Ω, A and Xα, α ∈ [−1, 1], be as in Examples 51.4 and q > n.Let f ∈ C1(R) and let F be the Nemytskii mapping associated with f , that isF (u)(x) = f(u(x)). Assume also that either f(0) = 0 or Ω is bounded. Fix β = 1/2,α = 1 and let u0 ∈ Xβ . Recall from Examples 51.4 that Xβ = W 1,q

0 (Ω) or Xβ =W 1,q(Ω) if we consider Dirichlet or Neumann boundary conditions, respectively.Since W 1,q(Ω) → L∞ ∩ Lq(Ω), we see that the assumptions of Theorem 51.7are satisfied and we obtain a unique maximal solution u ∈ C([0, Tmax), Xβ). Inaddition, u ∈ C1−γ((0, Tmax), Xγ) for γ ∈ [1/2, 1). Choose γ such that ρ := 1−γ =(1− n/q)/3. Then Xγ → BUC1+ρ(Ω), hence u ∈ Cρ((0, Tmax), BUC1+ρ(Ω)).

Remark 51.8(i) implies u + Aγ−1u = F (u) in (0, Tmax). Fix 0 < δ < T < Tmax,choose ψ ∈ C∞(R) such that ψ(t) = 0 for t ≤ δ/2, ψ(t) = 1 for t ≥ δ, and setv(t) := ψ(t)u(t). Then

v + Aγ−1v = f in (0, Tmax), v(0) = 0, (51.16)

where f(t) := ψ(t)F (u(t))+ψt(t)u(t). Assume that the coefficients of the operatorA belong to BUCρ(Ω) and Ω is a bounded domain of class C2+ρ. Since f is alsoHolder continuous, Theorem 48.2(ii) shows that there exists a classical solutionw of problem (51.16). The uniqueness of solutions of (51.16) guarantees w = v,hence u is a classical solution for t > 0. Theorem 48.2(ii) also implies

u ∈ BC2,1(Ω× [t1, t2]) whenever 0 < t1 < t2 < Tmax. (51.17)

If Ω is unbounded, then (51.17) can be shown by using a smooth cut-off functionin the x-variable.

This example can be straightforwardly modified for more general nonlinearitiesand systems (cf. also Remark 51.8(ii)). If F (t, u)(x) = f(x, t, u(x, t),∇u(x, t)), forexample, then one obtains the existence of a maximal solution u ∈ C([0, Tmax), Xβ)provided Ω is bounded, the function f = f(x, t, u, ξ) is C1, its derivatives satisfythe growth condition

|∂tf |+ |∂uf |+ (1 + |ξ|)|∂ξf | ≤ C(|u|)(1 + |ξ|p)and q > n max(1, p − 1) (see [14] for details). Note that the regularity of f withrespect to x and t can be considerably relaxed.

Example 51.10. Let Ω, A and Xα, α ∈ [−1, 1], be as in Example 51.4(i), and letp > 1 and

q > max(1,

n(p− 1)p + 1

,np

n + p

).

Fix z ∈ (max(1, q/p, nq/(n+q)), min(q, nq/p(n−q)+)] and assume that F (u)(x) =f(x, u(x)) where f ∈ C1 satisfies f(·, 0) ∈ Lz(Ω) and |∂uf(x, u)| ≤ a(x) + C|u|p−1

with a ∈ Lp′z(Ω) (the regularity of f with respect to x can be relaxed). Set

β =12

and α =12

(2 +

n

q− n

z

).

Then α ∈ (β, 1], Xβ = W 1,q0 (Ω) → Lpz(Ω) and Lz(Ω) → Xα−1 (due to X1−α(q′)

→ W 2−2α,q′(Ω) → Lz′

(Ω) and (51.7)). Since F considered as a map from Lpz(Ω)to Lz(Ω) is locally Lipschitz continuous (uniformly on bounded sets), it has thesame properties as a map F : Xβ → Xα−1. Consequently, given u0 ∈ W 1,q

0 (Ω),Theorem 51.7 guarantees the existence of a maximal solution u ∈ C([0, Tmax),W 1,q

0 (Ω)) satisfying u ∈ C((0, Tmax), Xγ) for all γ < α.Next assume f = f(u) and notice that this restriction and our assumptions on

f imply f(0) = f ′(0) = 0 if Ω is unbounded. In fact, if Ω is unbounded, thenits measure has to be infinite (since Ω is uniformly regular of class C2), hencethe spaces Lz(Ω) and Lp′z(Ω) do not contain nonzero constants. If q ≥ n orp ≤ n/(n− q), then we may set z = q, hence α = 1, and we obtain

u ∈ C((0, Tmax), Xγ) for all γ ∈ [1/2, 1). (51.18)

Assume q < n, p > n/(n − q), and consider t0 > 0 small and β ∈ (β, α). Sinceu(t0) ∈ Xβ we may repeat the considerations above with

z := min(q,

nq

p(n− 2βq)+

), α :=

12

(2 +

n

q− n

z

),

to obtain u ∈ C((0, Tmax), Xγ) for all γ < α. If q ≥ n/(2β) or p ≤ n/(n − 2βq),then z = q, α = 1 and we obtain (51.18) again. Otherwise we notice that z > z,α > α, and use a bootstrap argument (enlarging β, z and α) to see that (51.18) isalways true.

Next choose γ ∈ (β, 1) . Since

Xγ = W 2γ,q ∩W 1,q0 (Ω) →W 1,q

0 (Ω) for some q > q,

we can repeat the arguments above with q replaced by q. An obvious bootstrapw.r.t. q shows

u ∈ C((0, Tmax), W 2γ,q ∩W 1,q0 (Ω)) for all γ < 1 and q ∈ [q,∞).

476 Appendices

Notice also that the considerations in Example 51.9 guarantee now

u ∈ Cρ((0, Tmax), BUC1+ρ ∩W 1,q0 (Ω)) (51.19)

for some ρ ∈ (0, 1) and all q ∈ [q,∞).Fix t ∈ (0, Tmax). Then the bootstrap argument used above, Remark 51.8(iii)

and the embedding W 2γ,q(Ω) → BUC1(Ω) for suitable γ, q guarantee the existenceof ε, C > 0 (depending on t and max0≤s≤t ‖u(s)‖W 1,q(Ω)) such that given u0 ∈W 1,q

0 (Ω) satisfying ‖u0 − u0‖W 1,q(Ω) < ε, we have Tmax(u0) > t and

‖u(t)− u(t)‖BC1 < C‖u0 − u0‖W 1,q(Ω). (51.20)

In fact, Remark 51.26(iii) and Example 51.27 below guarantee that the RHS in(51.20) can be replaced with C‖u0−u0‖r for any r > n(p−1)/2, r > 1. In addition,estimate (15.18) shows that the same is true if r = 1 > n(p− 1)/2.

Next assume that f ′ is locally Holder continuous. Then (51.19) implies theexistence of ρ > 0 such that F (u) ∈ Cρ((0, Tmax), X1/2). Now Theorem 51.1(v)guarantees u ∈ C1+ρ((0, Tmax), X1/2) ∩ C((0, Tmax), X1), hence

u ∈ Wq := C1+ρ((0, Tmax), W1,q0 (Ω)) ∩ C((0, Tmax), W 2,q(Ω)).

Since u(t) ∈ W 1,q0 (Ω) for all t > 0 and q ≥ q the arguments above imply u ∈ Wq

for all q ∈ [q,∞).Next assume that f ′′ is locally Holder continuous. We will show that ut ∈ Wq

for all q ∈ [q,∞). Fix 0 < δ < Tmax/2 and choose a cut-off function ψ ∈ C∞(R)such that ψ(t) = 0 for t ≤ δ and ψ(t) = 1 for t ≥ 2δ. Notice that the function utψformally solves the linear problem

wt + Aw = f ′(u)utψ + utψt in Ω× (0, Tmax),

w = 0 on ∂Ω× (0, Tmax),

w(·, 0) = 0 in Ω.

(51.21)

Theorem 51.1(v) guarantees that the solution w of (51.21) belongs to Wq. SetW (t) :=

∫ t

0(w(s) + u(s)ψt(s)) ds. Then is is easy to see that both W and uψ

satisfy the same linear problemWt + AW = f(u)ψ + uψt in Ω× (0, Tmax),

W = 0 on ∂Ω× (0, Tmax),

W (·, 0) = 0 in Ω,

hence W ≡ uψ and Wt(s) = ut(s) for s > 2δ. Since Wt = w+uψt ∈ Wq and δ > 0was arbitrary, we see that ut ∈ Wq. If we only assumed that f ′ is locally Holdercontinuous (instead of f ′′ locally Holder continuous) and if the coefficients of Abelong to BUCρ(Ω) and Ω is of class C2+ρ, then applying Lp- and subsequentlySchauder estimates to (51.21) (along with a cut-off argument if Ω is unbounded)would yield ut ∈ C2+ρ,1+ρ/2(Ω × (0, T )) for suitable ρ > 0. Similar regularityproperties can be derived in the same way in the case of Neumann boundaryconditions.

Remark 51.11. Let Ω ⊂ Rn be uniformly regular of class C2, let X0 be any of the

spaces L∞(Ω), BC(Ω), BUC(Ω), C∗(Ω) := u ∈ BUC(Ω) : lim|x|→∞ u(x) = 0,and

Au = −n∑

i,j=1

aij∂2

∂xi∂xju +

n∑i=1

bi∂

∂xiu + cu,

where aij , bi, c ∈ BUC(Ω) and aij = aji are uniformly elliptic. Let A be theunbounded operator in X0 defined by Au = Au for u ∈ D(A), where

D(A) =

u ∈⋂q≥1

W 2,qloc (Ω) : u,A(u) ∈ X0, u = 0 on ∂Ω

.

Note that X0 is not reflexive and A is not densely defined, in general, since

D(A)X0 = u ∈ BUC(Ω) ∩X0 : u = 0 on ∂Ω.

Nevertheless, [344, Corollary 3.1.21] guarantees that −A is sectorial, hence it gener-ates an analytic semigroup e−tA in X0 (see [344] for the definition and properties ofsectorial operators). Notice that this semigroup is not strongly continuous if D(A)

is not dense in X0. However, its restriction to D(A)X0

is strongly continuous,cf. [344, Remark 2.1.5].

Let X1 := D(A) be endowed with the graph norm and let (Xγ , | · |γ), γ ∈ (0, 1),be Banach spaces satisfying

(X0, X1)γ,1 → Xγ → (X0, X1)γ,∞. (51.22)

We will also assume that the spaces Xγ have the following property: if Aγ denotesthe Xγ-realization of A, then

−Aγ is sectorial in Xγ and σ(Aγ) ⊂ σ(A), γ ∈ (0, 1). (51.23)

For example, if Xγ = (X0, X1)γ , where (·, ·)γ is any of the real interpolationfunctors (·, ·)γ,p, 1 ≤ p ≤ ∞, or the complex interpolation functor [·, ·]γ , then both(51.22) and (51.23) are true, see [344]. Similarly, if X0 = BC(Ω), then the space

X1/2 := u ∈ BC1(Ω) : u = 0 on ∂Ω (51.24)

satisfies (51.22), (51.23) with γ = 1/2 due to [344, Propositions 3.1.27, 3.1.28,Theorem 3.1.25] and standard elliptic regularity theory. In general, [344, Proposi-tion 3.1.28] and (51.22) imply

Xγ → BUC2γ−ε(Ω), γ ∈ (0, 1], 0 < ε < 2γ. (51.25)

The proofs of [344, Propositions 2.3.1, 2.2.9] show that the semigroup e−tA

satisfies estimates (51.3) for 0 ≤ β ≤ α < 1. These estimates can be used for

478 Appendices

the proof of similar existence results as above. To be more precise, assume that1 = α > β ≥ 0, u0 ∈ Xβ and F : Xβ → X0 is locally Lipschitz continuous,uniformly on bounded subsets of Xβ . Then [344, Theorem 7.1.2] guarantees theexistence of r, T > 0 such that, given u0 ∈ Xβ , |u0−u0|β < r, the integral equation(51.9) has a unique solution u ∈ L∞((0, T ), Xβ). In addition, u ∈ C([0, T ], Xδ) forδ ∈ [0, β),

(u− e−tAu0) ∈ C([0, T ], Xβ), (51.26)

andu ∈ C1−γ((0, T ], Xγ), γ ∈ (0, 1). (51.27)

These results imply the existence of a maximal solution and one can also provesimilar assertions to those in Remarks 51.8(ii)–(iv). Notice also that if β = 0 and

u0 ∈ D(A)X0 , then (51.26) and the strong continuity of the restriction of e−tA to

D(A)X0 guarantee u ∈ C([0, T ], X0).

In particular, if F (u)(x) = f(u(x)) with f ∈ C1, then F : X0 → X0 is locallyLipschitz continuous, uniformly on bounded sets. Therefore, setting α = 1 andβ = 0 we get a solution u of (51.9) on the maximal time interval [0, Tmax(u0)) forany u0 ∈ X0. In addition, the analogue of Remark 51.8(iii) and (51.25) guaranteethe following: if u0 ∈ L∞(Ω) and t ∈ (0, Tmax(u0)) are fixed, then there existC, ε > 0 such that Tmax(u0) > t and

‖u(t)− u(t)‖BC1 ≤ C‖u0 − u0‖∞ (51.28)

for any u0 ∈ L∞(Ω) satisfying ‖u0 − u0‖∞ < ε.If F (u)(x) = f(u(x),∇u(x)) with f ∈ C1 and X0 = BC(Ω), then F has obvi-

ously the required continuity properties as a map F : X1/2 → X0, where X1/2 isdefined in (51.24). Hence, setting α = 1 and β = 1/2 we get a maximal solution

u ∈ C([0, Tmax), X1/2) (51.29)

provided u0 ∈ X1/2. Of course, analogous statements are true for nonlinearities ofthe form F (u)(x) = f(x, u(x),∇u(x)) or F (t, u)(x) = f(x, t, u(x),∇u(x)), cf. Re-mark 51.8(ii) and [344].

Finally, similar results are true if we consider Neumann boundary conditionsinstead of Dirichlet boundary conditions (that is, if we replace the condition u = 0on ∂Ω in the definition of D(A) by ∂u/∂ν = 0 on ∂Ω) see [344, Corollary 3.1.24and Theorem 3.1.26], for example.

Example 51.12. Let Ω ⊂ Rn be uniformly regular of class C2, f, g ∈ C1, d1, d2 >

0 and consider the system

ut − d1∆u = f(u, v),

vt − d2∆v = g(u, v),

x ∈ Ω, t > 0, (51.30)

complemented with homogeneous Dirichlet boundary conditions if Ω = Rn. Con-

sider also initial data u0, v0 ∈ L∞(Ω). Set X0 = L∞ × L∞(Ω),

X1 =

(u, v) : u, v ∈⋂q≥1

W 2,qloc (Ω), u, v, ∆u, ∆v ∈ L∞(Ω), u|∂Ω = v|∂Ω = 0

,

Xγ = (X0, X1)γ , γ ∈ (0, 1), A(u, v) = (−d1∆u,−d2∆v) for (u, v) ∈ X1 andF (u, v) = (f(u, v), g(u, v)). Then F : X0 → X0 is locally Lipschitz continuous (uni-formly on bounded sets) and a straightforward modification of Remark 51.11 showsthat the problem has a unique maximal solution satisfying (u, v)− e−tA(u0, v0) ∈C([0, Tmax), X0) and (u, v) ∈ C1−γ((0, Tmax), Xγ) for any γ < 1. Using the ana-logue of (51.25) we see that both u and v solve linear scalar problems with Holdercontinuous right-hand sides so that one can use Schauder estimates to prove higherregularity of these solutions.

Analogous assertions as above are also true in the case of homogeneous Neu-mann conditions (or Dirichlet conditions for u and Neumann conditions for v).In addition, if we prescribe inhomogeneous Neumann boundary conditions of theform ∂νu = h1(t), ∂νv = h2(t), where h1, h2 are smooth, then we can find smoothfunctions uh, vh satisfying these boundary conditions and we obtain the existenceresults by solving the system

∂tu− d1∆u = f(u + uh, v + vh) + d1∆uh − ∂tuh,

∂tv − d2∆v = g(u + uh, v + vh) + d2∆vh − ∂tvh,

with homogeneous Neumann boundary conditions (by using the analogue of Re-mark 51.8(ii)).

Finally, using (the analogues of) Remarks 51.8(ii),(iv) one can also solve theproblem if the functions f, g are defined for nonnegative (or positive) argumentsonly, provided the initial data are nonnegative (or positive) and either f, g ≥ 0 orwe can guarantee the positivity of the solution by other means.

Example 51.13. Let Ω ⊂ Rn be a bounded domain of class C2+ρ for some

ν ∈ (0, 1), let f, g be C1-functions and consider the equation

ut −∆u = f(u,

∫Ω

g(u) dx), x ∈ Ω, t > 0, (51.31)

complemented with homogeneous Dirichlet boundary conditions. Assume also thatthe initial data u0 ∈ L∞(Ω). Since the nonlinearity

F : L∞(Ω)→ L∞(Ω) : u → f(u,

∫Ω

g(u) dx)

480 Appendices

is locally Lipschitz (uniformly on bounded sets), we can use Remark 51.11 in orderto solve the problem. Similarly as in Example 51.12, we can also consider Neu-mann boundary conditions and nonlinearities defined for nonnegative or positivearguments only, for example

F (u) = up(∫

Ω

uq dx)−m

, (51.32)

where p, q ≥ 1 and m > 0.The same arguments apply to the equation

ut −∆u =(∫

Rn

Kup)r

uq, x ∈ Rn, t > 0, (51.33)

where p, q ≥ 1, r > 0, K ∈ L1(Rn) is positive and continuous, the initial datau0 ∈ L∞(Rn) are nonnegative and not identically zero. If in addition u0 ∈ L1(Rn),then the assumption K ∈ L1(Rn) can be replaced by K ∈ L∞(Rn). In fact, theexistence of a unique mild solution u ∈ L∞((0, T ), X), X := L1 ∩L∞(Rn), followsby a direct application of the Banach fixed point theorem to the mapping definedin (51.13) in a ball of the space L∞((0, T ), X).

Further regularity of solutions of the above problems can be obtained by con-sidering those problems as linear problems with bounded (or Holder continuous)RHS, cf. Examples 51.9, 51.10. In particular, the solutions of (51.31) are classicalfor t > 0.

Finally, let us consider the homogeneous Neumann problem for the nonlinearity(51.32) with p = q > 1, m = 1 (see (44.24)). Assume that Ω is the unit balland u0 ∈ C2(Ω) is radial and positive, u0(x) = U0(|x|) where U ′

0(1) = 0. Thenu(x, t) = U(|x|, t) for some U = U(r, t). As mentioned above, u is a classicalsolution for t > 0. Set T := Tmax. Theorem 51.7 (with the choice α = 1, β = 1− ε,ε > 0 small, and X0 = Lr(Ω), r > n/(1− 2ε)) also guarantees

u ∈ C([0, τ ], W 2−2ε,r(Ω)) → C([0, τ ], C1(Ω)), 0 < τ < T.

The function v(x, t) := Ur(|x|, t) ∈ C(Ω × [0, T )) is a weak (and, consequentlystrong) solution of the linear equation vt −∆v = (

∫Ω

up)−1pup−1v complementedby homogeneous Dirichlet boundary conditions on ST . Now Schauder estimatesimply v ∈ C2,1(QT ).

Example 51.14. Let Ω ⊂ Rn be a bounded domain of class C2+ρ for some

ν ∈ (0, 1), let p > 1, q ≥ 1, k ≥ 0, and consider the problem

ut −∆u =∫ t

0

|u|p−1u(x, s) ds− k|u|q, x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω t > 0,

u(x, 0) = u0, x ∈ Ω.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (51.34)

First notice that if F (u) =∫ t

0F1(u(s)) ds + F2(u) where F1, F2 satisfy the

assumptions of Theorem 51.7, then a straightforward modification of the proofshows that the first part of that theorem remains true. More precisely, given u0 ∈Xβ there exists a unique local solution u ∈ C([0, T ], Xβ) and u ∈ C((0, T ], Xγ)for all γ ∈ [β, α), |u(t)|γ ≤ Ctβ−γ for t > 0. Combining these arguments withRemark 51.11 we see that problem (51.34) is well-posed in X0 := L∞(Ω) and thesolution satisfies u ∈ C((0, T ], Xγ) and |u(t)|γ ≤ Ct−γ for all γ ∈ [0, 1), where Xγ ,γ ∈ (0, 1], are the spaces from Remark 51.11.

Fix t0 ∈ (0, T ) and ε > 0 small. Since the nonlinearity F1(u) = |u|p−1u satisfies‖F1(u)‖BC1 ≤ C(‖u‖∞)‖∇u‖∞, u ∈ BC1(Ω) : u|∂Ω = 0 → X1/2−ε and (51.25)is true, we see that

|F1(u(s))|1/2−ε ≤ ‖F1(u)‖BC1 ≤ C|u(s)|1/2+ε ≤ Cs−1/2−ε

and, in particular,

F 0 :=∫ t0

0

F1(u(s)) ds ∈ X1/2−ε → BUC1−3ε(Ω). (51.35)

Parabolic Lp-estimates and embedding (1.2) guarantee that u is Holder continuousin both x and t for t ≥ t0, hence F (u)−F 0 is Holder continuous. Now (51.35) andSchauder estimates guarantee that u is a classical solution of (51.34) for t > 0.Obviously this remains true for the maximal solution on (0, Tmax). (Notice thatthe Holder continuity of u for t > 0 also follows from Remark 51.8(ii) with thechoice 1 > α > β > 0, r > max(1/(α− β), n/2β) and X0 = Lr(Ω).)

Finally assume k = 0. Set T := Tmax. Similar arguments as at the end ofExample 51.10 show that ut solves the linear problem

vt −∆v = |u|p−1u, x ∈ Ω, 0 < t < Tmax,

v = 0, x ∈ ∂Ω, 0 < t < Tmax,

(51.36)

hence Schauder estimates guarantee

v = ut ∈ C2,1(QT ) ∩ C(Ω× (0, T )).

Let ϕ ∈ C2(Ω× [0, T )), ϕ = 0 on ∂Ω× [0, T ). Multiplying the equation in (51.34)with ϕ, integrating by parts and passing to the limit as t→ 0+ we obtain

limt→0+

∫Ω

ut(x, t)ϕ(x, t) dx =∫

Ω

u0(x)∆ϕ(x, 0) dx.

In particular, if u0 ∈ H2 ∩H10 (Ω), then

limt→0+

∫Ω

v(x, t)ϕ(x, t) dx =∫

Ω

∆u0(x)ϕ(x, 0) dx.

Now we infer from the uniqueness proof in Proposition 48.9 that v is (a strong) so-lution of (51.36) with initial data ∆u0. In particular v = ut ∈ C([0, T ), L2(Ω)).

482 Appendices

Example 51.15. Consider the problem

ut −∆u = c|u|p−1u− a · ∇(|u|q−1u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (51.37)

where Ω ⊂ Rn is uniformly regular of class C2+ρ, ρ ∈ (0, 1), p > 1, q ≥ 1, c ≥ 0

and a ∈ Rn, a = 0.

Assume first that u0 ∈ W 1,r0 (Ω) with r ∈ (n,∞). Set X0 := Lr(Ω), X1 :=

W 2,r ∩W 1,r0 (Ω), Au := −∆u for u ∈ X1, and F (u) := |u|p−1u − a · ∇(|u|q−1u).

Let Xθ = Xθ(r) be defined as in Example 51.4(i), in particular X1/2 = W 1,r0 (Ω).

Choose ε > 0 such that (1−2ε)r > n and set α := 1/2 and β := 1/2−ε. Notice thatF satisfies the assumptions of Theorem 51.7 since F can be viewed as F = F1+F2,where

F1 : Xβ → Lpr(Ω)|u|p−1u−−−−−→ Lr(Ω) → Xα−1,

F2 : Xβ → Lqr(Ω)|u|q−1u−−−−−→ Lr(Ω) a·∇r

−−−→ (W 1,r′0 (Ω))′ = Xα−1,

and ∇r is defined by

〈∇rw, ϕ〉 := −∫

Ω

w∇ϕdx ϕ ∈W 1,r′0 (Ω), w ∈ Lr(Ω).

Consequently, (51.37) possesses a unique solution u ∈ C([0, T ], Xβ).Next consider the case

u0 ∈ X := u ∈ BC1(Ω) : u|∂Ω = 0. (51.38)

Set X0 := BC(Ω), Au := −∆u for u ∈ D(A), where D(A) (and Xθ, θ ∈ (0, 1])are as in Remark 51.11, X1/2 = X . If q ≥ 2 or q = 1, then the functionf(u, ξ) := |u|p−1u − q|u|q−1(a · ξ) is C1, hence F : X1/2 → X0 has the requiredcontinuity properties and (51.37) possesses a unique solution u ∈ L∞((0, T ), X1/2)satisfying u ∈ C([0, T ], Xδ) for δ < 1/2, (u−e−tAu0) ∈ C([0, T ], X1/2) and (51.27),see Remark 51.11. In addition, this solution can be continued on the maximal ex-istence interval [0, Tmax(u0)) and Remark 51.35 with the choice α = 1, γ = 0 andβ = 1/2 (or Proposition 51.34 with α = 1, γ = 0 and β ∈ (1/2, 1)) guarantee that

if Tmax(u0) <∞, then lim supt→Tmax(u0)

‖u(t)‖∞ = ∞. (51.39)

Finally, Schauder estimates show that u is a classical solution for t > 0 and themaximum principle guarantees that u ≥ 0 if u0 ≥ 0.

Obviously, all the assertions above concerning the case (51.38) remain truefor all q ≥ 1 if we replace the nonlinearity |u|q−1 in the definition of f with(|u| + ε)q−1, ε ∈ (0, 1]. Let uε denote the corresponding solution. Then the argu-ments in Remark 51.35 (with α = 1, γ = 0 and β = 1/2) guarantee that givenT < ∞, C0 > 0,

‖∇uε(t)‖∞ ≤ C1, t ∈ [0, T ], provided ‖uε(t)‖∞ ≤ C0, t ∈ [0, T ], (51.40)

where the constant C1 > 0 does not depend on ε. In the following proposition weuse the approximation solutions uε in order to show the solvability of (51.37) inX1/2 for any q ≥ 1. For simplicity we restrict ourselves to nonnegative solutionsand to the case Ω bounded or Ω = R

n.

Proposition 51.16. Let Ω ⊂ Rn be a bounded domain of class C2+γ for some

γ ∈ (0, 1) or Ω = Rn. Consider the problem (51.37), where

u0 ∈ X+ = u ∈ BC1(Ω) : u|∂Ω = 0, u ≥ 0.

(i) There exists a unique, maximal classical solution u ∈ C2,1(Ω×(0, T )) of (51.37),such that u ∈ C([0, T ), C(Ω)) (u ∈ C([0, T ), BC(Rn)) if Ω = R

n) and ∇u ∈L∞

loc([0, T ), L∞(Ω)). Moreover, (51.39) is true (with Tmax(u0) = T ).(ii) Let Ω = R

n. Then u also satisfies

u ∈ L∞loc((0, T ), BC2(Rn)). (51.41)

If in addition u0 ∈ L1(Rn), then

u ∈ C([0, T ), L1(Rn)). (51.42)

Proof. The uniqueness of the solution is guaranteed by the comparison principlein Proposition 52.16.

To establish existence, we consider the approximating problem

∂tuε −∆uε = cupε − q(uε + ε)q−1(a · ∇uε), x ∈ Ω, t > 0,

uε = 0, x ∈ ∂Ω, t > 0,

uε(x, 0) = u0(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (51.43)

By Example 51.15, for each ε ∈ (0, 1], there exists Tε ∈ (0, 1] and a unique classicalsolution uε ∈ L∞([0, Tε], X+) of (51.43) satisfying (uε − e−tAu0) ∈ C([0, Tε], X+).Moreover, uε can be continued as long as ‖uε(t)‖∞ remains bounded and (51.40) istrue. By comparing with the solution of the ODE y′(t) = cyp, y(0) = M := ‖u0‖∞,we see that

0 ≤ uε(t) ≤ 2M, 0 < t ≤ T0 := C(p)M1−p. (51.44)

484 Appendices

In particular, Tε ≥ T0 and

‖∇uε(t)‖∞ ≤ C1, 0 < t ≤ T0. (51.45)

Now, (51.45) guarantees that the RHS of (51.43) is bounded in L∞(QT0) inde-pendently of ε. In the case Ω bounded, by parabolic Lr-estimates and the embed-ding (1.2), it follows that uε is bounded in C1+σ,σ/2(Ω×(0, T0]) for some σ ∈ (0, 1).Applying this estimate to the RHS we deduce from Schauder estimates that uε isbounded in C2+α,1+α/2(Ω × (0, T0]) for some α ∈ (0, 1). Therefore (some subse-quence of) uε converges to a classical solution u ∈ C2,1(Ω×(0, T0]) of (34.4). In thecase Ω = R

n, we may apply the same argument in B(x0, 1) for each x0 ∈ Rn and we

obtain a bound of uε in C2+α,1+α/2(B(x0, 1)× (0, T0]), with constant independentof x0. This yields a solution u ∈ C2,1(Rn× (0, T0]) of (34.4), with ut, D

2u boundedin R

n × (τ, T0] for each τ > 0. Note that this implies u ∈ C((0, T0], BC(Rn)) and(51.41). Moreover, in both cases, (51.45) and the variation-of-constants formulaimply

‖uε(t)− e−tAu0‖∞ ≤ Ct(Mp + (M + 1)q−1C1

). (51.46)

Passing to the limit ε → 0, we get (51.46) with u(t) instead of uε(t), hence thecontinuity of u(t) in C(Ω) (or in BC(Rn)) at t = 0.

Since the solution u satisfies the variation-of-constants formula for t > 0, asser-tion (51.39) follows from Proposition 51.34 or Remark 51.35 (cf. the same argumentin Example 51.15).

Finally, let us prove (51.42). Observe that for Ω = Rn and u ∈ C2,1(Rn×(0, T ))

such that u ∈ C([0, T ), BC(Rn))∩L∞((0, T ), W 1,∞(Rn)), (51.37) is equivalent tothe integral equation

u(t) = Gt ∗ u0 + c

∫ t

0

Gt−s ∗ up(s) ds +∫ t

0

(a · ∇Gt−s) ∗ uq(s) ds. (51.47)

When u0 ∈ X+ ∩ L1(Rn), one can obtain a (unique) solution of (51.47) as a fixedpoint in a suitable ball of the metric space C([0, T ], L1 ∩ BC(Rn)) ∩ L∞((0, T ),W 1,∞(Rn)) endowed with its natural norm. This can be done by using similar (andin fact simpler) arguments as in the proof of Theorem 15.3, using in particular thefact that ‖∇Gt ∗ f‖1 ≤ Ct−1/2‖f‖1, f ∈ L1(Rn). Moreover, it is easy to see thatthe solution of (51.47) can be continued in this space as long as ‖u(t)‖∞ remainsbounded. By the already proved uniqueness statement, we deduce that the twosolutions coincide, with same existence time, which completes the proof.

51.3. Stability of equilibria

Theorem 51.17. Let α, β, F be as in Theorem 51.7. Let, in addition, ω(−A) < 0and

|F (u)|α−1 = o(|u|β) as |u|β → 0.

Then the zero solution of (51.9) is (locally) exponentially asymptotically stable.More precisely, given ω ∈ (ω(−A), 0) there exist δ∗ > 0 and C > 0 such that thesolution u with initial data u0 satisfying |u0|β < δ∗ exists globally and

|u(t)|β ≤ Ceωt|u0|β for all t ≥ 0. (51.48)

Proof. Let ω ∈ (ω(−A), 0). Choose ω ∈ (ω(−A), ω). Then (51.3) guarantees

‖e−tA‖L(Xα−1,Xβ) ≤ C(ω, A)tα−1−βeωt,

‖e−tA‖L(Xβ ,Xβ) ≤ C(ω, A)eωt,

for all t > 0, (51.49)

where C(ω, A) ≥ 1. Set

C∗ = C(ω, A)∫ ∞

0

τα−1−βe(ω−ω)τ dτ

and choose ε > 0 such that

|F (u)|α−1 ≤1

2C∗ |u|β whenever |u|β ≤ ε. (51.50)

Choose δ∗ = ε/2C(ω, A) and let |u0|β < δ∗. We may assume u0 = 0. Set

T = supt ∈ (0, Tmax(u0)) : |u(s)|β ≤ 2C(ω, A)eωs|u0|β for all s ∈ [0, t]

and notice that T > 0 and |u(s)|β ≤ ε for all s ∈ [0, T ). If T = ∞, then (51.48)is true. Hence, assume T < ∞. Then T < Tmax(u0) due to the uniform bound of|u(s)|β for s ∈ [0, T ), hence

|u(T )|β = 2C(ω, A)eωT |u0|β . (51.51)

On the other hand, using (51.49), (51.50), the inequality in the definition of T andthe definition of C∗ we obtain

|u(T )|β ≤ C(ω, A)eωT |u0|β + C(ω, A)∫ T

0

(T − s)α−1−βeω(T−s)|F (u(s))|α−1 ds

≤ C(ω, A)eωT |u0|β +C(ω, A)2

C∗ eωT |u0|β∫ T

0

(T − s)α−1−βe(ω−ω)(T−s) ds

< C(ω, A)eωT |u0|β + C(ω, A)eωT |u0|β ≤ 2C(ω, A)eωT |u0|β ,

which contradicts (51.51) and concludes the proof.

486 Appendices

Remarks 51.18. (i) A combination of Theorem 51.17 with estimates of the form(51.10) or (51.20) shows that the solution u in Theorem 51.17 also decays expo-nentially to zero in stronger norms than | · |β.

(ii) Theorem 51.17 can also be used in order to prove the stability of a non-zeroequilibria. In fact, let w ∈ Xα, Aα−1w = F (w). Assume that F : Xβ → Xα−1 isFrechet differentiable at w and set

Fw(v) := F (w + v)− F (w) − F ′(w)v,

hence|Fw(v)|α−1 = o(|v|β) as |v|β → 0.

Let us first consider the special case α = 1. Set X1 = X1, X0 = X0, A =A− F ′(w) (with domain X1) and assume that

A generates a C0 analytic semigroup in X0 and ω(−A) < 0. (51.52)

Notice that if A has compact resolvent, then F ′(w) ∈ L(Xβ , X0) is a compactperturbation of A, hence the first part of (51.52) is automatically satisfied. Setv(t) = u(t)− w and v0 = u0 − w. Then (51.9) can be written as

v(t) = e−tAv0 +∫ t

0

e−(t−s)AFw(v(s)) ds

and one can use Theorem 51.17 with A and F replaced by A and Fw, respectively.If α < 1 and F ′(w)|X1 ∈ L(X1, X0), then one can still use the same arguments as

above (provided (51.52) is true). In the general case we set X0 = Xα−1, X1 = Xα,A = Aα−1 − F ′(w) (with domain X1) and assume that (51.52) is true. Set alsoα = 1 and choose β ∈ (β+1−α, 1). Then Xβ → Xβ and one can use Theorem 51.17with A, F, α, β replaced by A, Fw, α, β, respectively.

(iii) The conclusions of Theorem 51.17 remain true in the situation of Remark51.11.

Theorem 51.19. Let α, β, F be as in Theorem 51.7, p > 1 and

|F (u)|α−1 = O(|u|pβ) as |u|β → 0. (51.53)

Assume that σ(−A) = ω1 ∪ σ2, where ω1 < 0 is a simple eigenvalue of −Awith eigenspace E1 and σ2 ⊂ λ : Reλ ≤ ω2 for some ω2 < ω1. Fix ω ∈(max(ω2, ω1p), ω1). Then there exist δ, C > 0 and a continuous map K : Xβ → E1

such that the solution u with initial data u0 satisfying |u0|β < δ exists globally and

|u(t)−K(u0)eω1t|β ≤ Ceωt|u0|β for all t ≥ 0. (51.54)

Proof. Let P1 and P2 denote the spectral projections in Xβ corresponding to thespectral sets ω1 and σ2, respectively, and E2 := P2(Xβ). Then Xβ = E1 ⊕ E2,E2 is A (and e−tA) invariant, σ(−A|E2) = σ2 and −A|E2 generates the analyticsemigroup e−tA|E2 (see [273, Section 1.5] and the references therein), hence (51.3)implies

|e−tAP2u|β ≤ Ceωt min(|u|β, tα−1−β |u|α−1) (51.55)

for all u ∈ Xβ . Choose ω ∈ (ω1, ω/p). Then Theorem 51.17 guarantees the exis-tence of δ ∈ (0, 1) and C > 0 such that

|u(t)|β ≤ Ceωt|u0|β (51.56)

whenever |u0|β < δ. Assume |u0|β < δ and denote ui(t) = Piu(t), i = 1, 2. SincePie

−tA = e−tAPi we have

u2(t) = e−tAP2u0 +∫ t

0

e−(t−s)AP2F (u(s)) ds,

hence (51.53), (51.55), (51.56) and ωp < ω imply

|u2(t)|β ≤ Ceωt|u0|β + C

∫ t

0

eω(t−s)(t− s)α−1−β |u(s)|pβ ds ≤ Ceωt|u0|β. (51.57)

Set

K(u0) := limt→∞u1(t)e−ω1t = P1u0 +

∫ ∞

0

e−ω1sP1F (u(s)) ds

(the integral is convergent since ‖P1F (u(s))‖E1 ≤ Ceωps|u0|pβ due to (51.53),(51.56) and ωp < ω < ω1). Now the assertion follows from (51.57) and the es-timate

‖u1(t)−K(u0)eω1t‖E1 = eω1t∥∥∥ ∫ ∞

t

e−ω1sP1F (u(s)) ds∥∥∥

E1

≤ Ceωpt|u0|pβ .

Remarks 51.20. (i) The proof of Theorem 51.19 implies

K(u0) = P1u0 + O(|u0|pβ).

(ii) Let us consider the situation from Remark 51.11 with Ω bounded, Au =−∆u, X0 = BC(Ω) and X1/2 defined by (51.24). Set α = 1 and β = 1/2. Thenthe statement in Theorem 51.19 remains true for this choice of spaces since A hasthe required properties, ω1 is a simple eigenvalue of A1/2 and σ(A1/2) ⊂ σ(A).

488 Appendices

51.4. Self-adjoint generators with compact resolvent

The proof of Theorem 51.21 below is based on an idea used in the construc-tion of stable manifolds for general semilinear parabolic problems (see [273, The-orem 5.2.1] or [101, Lemma 4.1], for example). We will use this idea in a specificsituation in order to obtain more precise information than that in [273] or [101].In addition to (51.2) we will also assume that

X0 is a Hilbert space,A : X0 → X0 is self-adjoint and has compact resolvent,ω1 > ω2 > · · · are all distinct eigenvalues of −A,

(·, ·)θ is the complex interpolation functor for all θ ∈ [0, 1].

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (51.58)

Then Xα, α ∈ [−1, 1], are Hilbert spaces and the operators Aα are self-adjoint(see [16, Theorem V.1.5.15]). Let Pi, Qi and Ri, i = 1, 2, . . . , denote the spectralprojections in X0 corresponding to the spectral sets ωi, ωi+1, . . . , ω1, . . . , ωi−1and ωi, respectively. Let Pi,α denote the restriction Pi|Xα if α > 0 and the closureof Pi in Xα if α < 0. Then Pi,α is the spectral projection in Xα corresponding tothe spectral set ωi, ωi+1, . . . and analogous assertions are true for Qi and Ri.Without fear of confusion we will write Pi, Qi, Ri instead of Pi,α, Qi,α, Ri,α. Since

−A =∞∑

j=1

ωjRj and e−tA =∞∑

j=1

eωjtRj ,

it is easy to see that there exist Ci > 0, i = 1, 2, . . . , such that

‖e−tAPi‖L(Xα,Xα) ≤ eωit, α ∈ [−1, 1], t ≥ 0,

‖e−tAPi‖L(Xα−1,Xα) ≤Ci

teωit, α ∈ [0, 1], t > 0,

and, by interpolation,

‖e−tAPi‖L(Xβ ,Xα) ≤ Citβ−αeωit, −1 ≤ β ≤ α ≤ 1, t > 0. (51.59)

Similarly,

‖e−tAQi‖L(Xβ ,Xα) ≤ Cieω1t, β, α ∈ [−1, 1], t ≥ 0,

‖etAQi‖L(Xβ ,Xα) ≤ Cie−ωi−1t, β, α ∈ [−1, 1], t ≥ 0,

(51.60)

where etAQi :=∑i−1

j=1 e−ωjtRj if t ≥ 0.

Theorem 51.21. Assume (51.58). Let α, β, F be as in Theorem 51.7, p > 1,F (0) = 0 and

|F (u)− F (v)|α−1 ≤ CF |u− v|β(|u|p−1

β + |v|p−1β

)for |u|β, |v|β small. (51.61)

Fix i ≥ 1 with ωi < 0 and choose λ ∈ [ωi, 0], λ < ωi−1 if i > 1. Then there existρ = ρi > 0 small and Ci > 0 with the following properties: given v0 ∈ PiXβ,|v0|β ≤ ρ, there exists a unique z0 ∈ QiXβ such that the solution of (51.9) withu0 := v0 + z0 is global and satisfies |u(t)|β ≤ 2ρeλt for all t ≥ 0. In addition,

|u(t)|β ≤ 2|v0|βeωit for all t ≥ 0 (51.62)

and|z0|β ≤ Ci|v0|pβ. (51.63)

Finally, if |Riv0|β > Ci|v0|pβ, then there exists c = c(v0) > 0 such that

|u(t)|β ≥ c|Riv0|βeωit for all t ≥ 0. (51.64)

Proof. Let u be a global solution of (51.9). Then u can be written in the formu = v + z, where

v(t) = e−tAv0 +∫ t

0

e−(t−s)APiF (u(s)) ds,

z(t) = e−tAz0 +∫ t

0

e−(t−s)AQiF (u(s)) ds,

(51.65)

v0 = Piu0, z0 = Qiu0 (z0 = 0 and z = 0 if i = 1). Assume first that

|u(t)|β ≤ ceλt, t ≥ 0, (51.66)

where c > 0 is small. If i > 1, then

|etAz(t)|β = |etAQiz(t)|β ≤ e−ωi−1t|z(t)|β ≤ ce(λ−ωi−1)t → 0 as t→∞,

|esAQiF (u(s))|β ≤ Cie−ωi−1s|F (u(s))|α−1 ≤ CiCF cpe(pλ−ωi−1)s,

(51.67)hence (51.65) guarantees

z0 = −∫ ∞

0

esAQiF (u(s)) ds (51.68)

and u = Φv0(u), where

Φv0(u)(t) := e−tAv0 +∫ t

0

e−(t−s)APiF (u(s)) ds−∫ ∞

t

e−(t−s)AQiF (u(s)) ds.

(51.69)

490 Appendices

On the other hand, if u is any function in C([0,∞), Xβ) satisfying (51.66) andu = Φv0(u) for some v0 ∈ PiXβ , then obviously u solves (51.9), where u0 = v0 +z0

and z0 is given by (51.68).Denote

‖u‖ = supt≥0

|u(t)|βe−λt

and Bρ = u ∈ C([0,∞), Xβ) : ‖u‖ ≤ 2ρ. We will show that, given v0 ∈ PiXβ ,|v0|β ≤ ρ, the mapping Φv0 possesses a unique fixed point in Bρ provided ρ > 0 issmall enough. Given u ∈ Bρ, we have

e−λt|Φv0(u)(t)|β ≤ e−λt‖e−tAPi‖L(Xβ ,Xβ)|v0|β

+∫ t

0

e−λt‖e−(t−s)APi‖L(Xα−1,Xβ)|F (u(s))|α−1 ds

+∫ ∞

t

e−λt‖e−(t−s)AQi‖L(Xα−1,Xβ)|F (u(s))|α−1 ds

≤ |v0|β + CiCF (2ρ)p[∫ t

0

(t− s)α−1−βe−(λ−ωi)(t−s)+λ(p−1)s ds

+∫ ∞

t

e(ωi−1−λ)(t−s)+λ(p−1)s ds]

< 2ρ,

provided ρ is small enough and i > 1 (analogous estimates are true for i = 1).Notice that the above estimates also imply

e−λt|Φv0(u)(t)− e−tAv0|β ≤ Ciρp

for some Ci > 0 and that similar estimates guarantee

‖Φv0(u)− Φv0(u)‖ ≤ 12‖u− u‖ for u, u ∈ Bρ.

Consequently, Φv0 : Bρ → Bρ is a contraction and possesses a unique fixed pointin Bρ. In addition, (51.68) and (51.67) imply |z0|β ≤ Ciρ

p. Repeating the abovearguments with ρ := |v0|β and λ := ωi we obtain estimates (51.62), (51.63) and

e−ωit|u(t)− e−tAv0|β ≤ Ci|v0|pβ . (51.70)

Set w0 := Riv0, y0 := Pi+1v0 = v0 − w0 and assume |w0|β > Ci|v0|pβ . Then

e−ωit|e−tAy0|β ≤ e−(ωi−ωi+1)t|y0|β ,

e−ωit|e−tAw0|β = |w0|β ,

hence (51.70) yields

e−ωit|u(t)|β ≥ |w0|β − e−(ωi−ωi+1)t|y0|β − Ci|v0|pβ > c|w0|β ,

provided c < 1−Ci|v0|pβ/|w0|β and t is large enough. This concludes the proof.

Corollary 51.22. Assume (51.58). Let α, β, F, p be as in Theorem 51.21, and letu be a global solution of (51.9) satisfying |u(t)|β → 0 as t→∞. Set

Λ := infλ ≤ 0 : limt→∞ |u(t)|βe−λt = 0

and assume Λ ∈ (−∞, 0). Then there exist C1, C2 > 0 and i ≥ 1 such that Λ = ωi

andC1e

ωit ≤ |u(t)|β ≤ C2eωit, t ≥ 0.

Proof. The same arguments as in the proof of [2, Corollary A.11] guarantee theexistence of i such that Λ = ωi, |u(t)|1/t

β → eωi and distXβ

(u(t)/|u(t)|β , Sβ

)→ 0 as

t →∞, where Sβ := v ∈ RiXβ : |v|β = 1. Choose λ ∈ (ωi, 0), λ < ωi−1 if i > 1,and let ρ = ρi > 0 be the constant from Theorem 51.21. Then |u(t)|1/t

β → eωi

implies |u(t + t0)|β ≤ ρeλt for t0 ≥ 0 large enough and all t ≥ 0. Enlarging t0if necessary we may assume |Riu(t0)|β > Ci|v0|pβ , where v0 := Piu(t0). Now theassertion follows from Theorem 51.21.

Remark 51.23. If 0 /∈ σ(A), then the assumption Λ < 0 in Corollary 51.22 isautomatically satisfied. In fact, using Theorem 51.21 with λ = 0 we obtain Λ ≤ ωi,where ωi is the largest negative eigenvalue of −A.

The assumption Λ > −∞ can be verified by an argument guaranteeing back-ward uniqueness (see [2, Lemma A.16 and Lemma B.4], for example, and cf. Ex-ample 51.24 below).

Example 51.24. Let L be the positive self-adjoint operator in the weighted spaceL2

g defined by (47.16). Recall that the domain of definition of L equals H2g , and L

has compact inverse. Consider the problem

vt + Av = |v|p−1v, y ∈ Rn, t > 0,

v(y, 0) = v0(y), y ∈ Rn,

(51.71)

where Av = Lv−λv and p > 1. Since L is self-adjoint and positive, it has boundedimaginary powers and −A generates a strongly continuous analytic semigroup inX0 := L2

g (see [16]). In addition, the domain of definition of A equals X1 := H2g .

Set Xθ = [X0, X1]θ for θ ∈ (0, 1) and X−1+θ = [X−1, X0]θ, where X−1 is thecompletion of X0 endowed with the norm |v|−1 = |L−1v|0. Then the abstractresults in [16] imply X−θ

.= X ′θ for θ ∈ (0, 1]. One can also easily verify X1/2 =

D(L1/2) = H1g (cf. Remark 51.2(i)).

Let p < pS . For simplicity assume n ≥ 3 (the case n ≤ 2 is similar). Then H1g →

L2∗g ∩L2

g due to Lemma 47.11 and, by interpolation, Xθ → Lrg ∩L2

g for θ ∈ [0, 1/2]and 1/r = 2θ/2∗+(1−2θ)/2. Using these embeddings, setting F (v) = |v|p−1v, z =min(2, 2∗/p), β = 1/2 and α = 1+(n/z′−n/2)/2 (cf. Example 51.10) one obtains

492 Appendices

that F : Xβ → Xα−1 and A satisfy the assumptions of Theorem 51.21. (Noticethat we could also choose α = 1 and β close to 1, β < 1, due to Remark 47.12(ii):in this case the assumption p < pS could be replaced by p(n− 4) < n.)

Now assume λ /∈ σ(L) and let v be a global solution of (51.71) satisfying|v(t)|β → 0 as t → ∞, v0 ≡ 0 (such solutions exist due to Theorem 51.21).We will show the following:(i) There exist C1, C2 > 0 and ωi ∈ σ(−A), ωi < 0, such that

C1eωit ≤ ‖v(t)‖∞ ≤ C2e

ωit, t ≥ 1. (51.72)

(ii) Assume ω1 < 0 and let φ1(y) = e−|y|2/4 be the corresponding eigenfunctionof A (see Lemma 47.13). Set ω := max(ω2, pω1) < ω1. Then there exists M =M(v0) ∈ R such that

‖v(t)−Meω1tφ1‖∞ ≤ Ceωt, t ≥ 0. (51.73)

Proof of (i). Set

Λ := infκ ≤ 0 : limt→∞ |v(t)|βe−κt = 0.

Then Λ < 0 due to Remark 51.23. Let us show that Λ > −∞.Choose κ < 0 and assume

|v(t)|β ≤ Ceκt, t ≥ 0. (51.74)

Let us prove that, given t0 > 0, estimate (51.74) and p < pS guarantee

‖v(t)‖∞ ≤ C(t0)eκt, t ≥ t0. (51.75)

Let r > 1, X0 = X0(r) = Lr(Rn), X1 = X1(r) = W 2,r(Rn) and let us rewrite(51.71) in the form

vt + Av = F (v), t > 0,

v(0) = v0,(51.76)

where Av = −∆v−λv is considered as an unbounded operator in X0 with domainX1 and F (v)(y) = |v(y)|p−1v(y) + (y · ∇v(y))/2. Let Xα, α ∈ [−1, 1] be the scaleof spaces constructed as in Example 51.4(i) (hence Xα → W 2α,r(Rn) for α ≥ 0),and let | · |∼α denote the norm in this space.

Since Xβ = X1/2 = H1g and

‖y · ∇v(t)‖2 ≤ C|v(t)|β ,

‖y · ∇v(t)‖1 ≤∫

Rn

|y| · |∇v(t)| dy ≤ C

∫Rn

(|∇v(t)g1/2)g−1/4 dy ≤ C|v(t)|β ,

‖|v(t)|p−1v(t)‖r = ‖v(t)‖ppr ≤ C|v(t)|pβ , r ∈ [max(1, 2/p), 2∗/p],

(51.74) guarantees‖F (v(t))‖r ≤ Ceκt, t ≥ τ, (51.77)

where r = min(2, 2∗/p) and τ = 0. Similar estimates as above and Lemma 47.11imply H1

g → L1 ∩ L2∗(Rn), hence H1

g → X0(r) for any r ∈ (1, 2∗].Fix ε, T > 0 small and assume that (51.77) is true for some r ∈ [min(2, 2∗/p), 2∗]

and τ ≥ 0. Then using estimates (51.3) (with A replaced by A) we infer for allt ≥ τ ,

|v(t + T )|∼1−ε ≤ C‖e−TA‖L(X0,X1−ε)|v(t)|∼0

+∫ t+T

t

‖e−(t+T−s)A‖L(X0,X1−ε)|F (v(s))|∼0 ds

≤ C|v(t)|β + C

∫ t+T

t

(t + T − s)ε−1eκs ds ≤ Ceκ(t+T ),

where C depends on T, ε and the constants C in (51.74) and (51.77). If r > n/2,then X1−ε → L∞(Rn) for ε small enough and (51.75) follows. If r ≤ n/2, thenX1−ε → Lq1 ∩W 1,q(Rn), where 1/q1 = 1/r− (2− 2ε)/n, 1/q = 1/r− (1− 2ε)/n,hence

‖v(t)‖q1 + ‖∇v(t)‖q ≤ Ceκt for t ≥ τ + T.

Notice that choosing ε small enough we have q1/p > r (due to p < pS and r ≥min(2, 2∗/p)), q := ε + (2− ε)q/2 > r and

‖|v(t)|p−1v(t)‖q1/p = ‖v(t)‖pq1≤ Cepκt,

‖y · ∇v(t)‖qq ≤ C

∫Rn

|∇v(t)|q−ε|∇v(t)|εgε/2 dy

≤ C‖∇v(t)‖q−εq ‖v(t)‖ε

H1g≤ Ceqκt,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ t ≥ τ + T,

hence‖F (v(t))‖r ≤ Ceκt, t ≥ τ ,

where τ = τ + T and r = min(q1/p, q) > r. An obvious bootstrap argumentconcludes the proof of (51.75).

Notice that similar estimates as above (or the choice α = 1 mentioned above)imply F (v) = |v|p−1v ∈ C((0,∞), X0). Consequently, Theorem 51.1(v) (used withX0 := X−ε, ε > 0 small) guarantees u ∈ C1((0,∞), X−ε) ∩ C((0,∞), X1−ε). Wealso have

|F (v)(t)|−ε ≤ C|F (v)(t)|0 ≤ Ch(t)|v(t)|0 ≤ Ch(t)|v(t)|1/2−ε, t ≥ t0,

whereh(t) := ‖|v|p−1(t)‖∞ ≤ C(t0)p−1e(p−1)κt

494 Appendices

belongs to L2(t0,∞), hence [2, Lemma A.16] yields

|v(t)|β ≥ c|v(t)|−ε ≥ cC1|v(t0)|−εe−C2(t−t0)

for suitable c, C1, C2 > 0. Consequently, Λ > −∞.Now we infer from Corollary 51.22 the existence of C1, C2 > 0 and ωi ∈ σ(−A),

ωi < 0, such thatC1e

ωit ≤ |v(t)|β ≤ C2eωit, t ≥ 0. (51.78)

Since (51.74) implies (51.75), we have

‖v(t)‖∞ ≤ C2eωit, t ≥ 1, (51.79)

and simple estimates based on the variation-of-constants formula also yield

|v(t)|β+ε ≤ C2eωit, t ≥ 1, (51.80)

where ε > 0 is small. Since Xβ+ε is compactly embedded into Xβ , given δ > 0 wecan find Cδ > 0 such that

|v(t)|β ≤ δ|v(t)|β+ε + Cδ‖v(t)‖∞. (51.81)

Choosing δ < C1/C2, estimates (51.78), (51.80) and (51.81) imply

‖v(t)‖∞ ≥ C1eωit, t ≥ 1,

for suitable C1 > 0. Consequently, (51.72) is true.Proof of (ii). Similarly as in (i), it suffices to prove (51.73) with ‖ · ‖∞ replaced

by | · |β. The proof of this modified estimate is almost the same as the proof of The-orem 51.19; one just needs to replace estimate (51.55) by the more precise estimate(51.59) and estimate (51.56) by |v(t)|β ≤ Ceω1t (which follows from the proof of(i) or from Theorem 51.19). The only difference appears in the case ω = pω1 = ω2,where one has to use a more precise estimate on the term e−(t−s)AP2F (v(s)).In fact, in this case Theorem 51.19 guarantees v(s) = Meω1sφ1 + w(s), where|w(s)|β ≤ Ce(ω2+ε)s and ε ∈ (0, ω1 −ω2) is such that δ := (p− 1)ω1 + ε < 0. Con-sequently, F (v(s)) = |M |p−1Mepω1sφp

1 + z(s), where |z(s)|α−1 ≤ Ce(ω2+δ)s. LetP3, R2 be the spectral projections introduced in the beginning of this subsection.Then R2φ

p1 = 0 due to Lemma 47.13, hence P2F (v(s)) = P3(|M |p−1Mepω1sφp

1) +P2z(s) and

|e−(t−s)AP2F (v(s))|β ≤ C(t− s)α−1−β[eω3(t−s)epω1s + eω2(t−s)e(ω2+δ)s

]= C(t− s)α−1−βeω2t

[e(ω3−ω2)(t−s) + eδs

].

This estimate is sufficient for the proof of (51.73) in the case ω = pω1 = ω2.

51.5. Singular initial data

In what follows we consider nonlinearities F : Xβ → Xα−1 with (at most) poly-nomial growth and we will show that under suitable assumptions one can ob-tain existence for initial data u0 ∈ Xδ with δ < β. The following theorem is anabstract analogue of Theorem 15.2. In addition, it also covers the critical case(cf. Remark 15.4(i)).

We assume that β > δ, M, T > 0, and we define the Banach space

YT := u ∈ L∞loc((0, T ], Xβ) : ‖u‖YT := sup

t∈(0,T )

tβ−δ|u(t)|β < ∞.

We also denote by BM = BM,T the closed ball in YT with center 0 and radius M .

Theorem 51.25. Assume that p > 1, 1 ≥ α > β > δ ≥ −1, δ > β − 1/p,β ≥ α− 1 and F : Xβ → Xα−1 satisfies

|F (u)− F (v)|α−1 ≤ CF |u− v|β(1 + |u|p−1

β + |v|p−1β

). (51.82)

Let u0 ∈ Xδ and let Φu0 be defined by (51.13).(i) If α > (β − δ)p + δ, then there exist M = M(|u0|δ) ≥ 1 and T = T (|u0|δ) > 0such that Φu0 possesses a unique fixed point in BM,T .(ii) If α = (β − δ)p + δ, then there exist M = M(u0) > 0 and T = T (u0) > 0 suchthat Φu0 possesses a unique fixed point in BM,T .

In both cases, u ∈ C([0, T ], Xδ) ∩ C((0, T ], Xγ) for any γ ∈ [δ, α).

Proof. We will use the Banach fixed point theorem for the mapping Φu0 : BM →BM . Increasing CF if necessary we may assume

|F (u)|α−1 ≤ CF (1 + |u|pβ). (51.83)

Let γ ∈ [δ, α), 0 < t ≤ T ≤ 1, M > 0, u ∈ BM , let CA be the constant from(51.11) and denote ξ+ := max(ξ, 0). Then

tγ−δ|Φu0(u)(t)|γ ≤ tγ−δ|e−tAu0|γ + tγ−δ∣∣∣ ∫ t

0

e−(t−s)AF (u(s)) ds∣∣∣γ

≤ CA|u0|δ + CACF tγ−δ

∫ t

0

(t− s)−(γ+1−α)+(1 + |u(s)|pβ

)ds

≤ CA|u0|δ + CACF tγ−δ

∫ t

0

(t− s)−(γ+1−α)+(1 + Mps(δ−β)p

)ds <∞

(51.84)

hence Φu0(u)(t) ∈ Xγ .

496 Appendices

(i) Let α > (β − δ)p + δ. Fix M ≥ max(1, 2CA|u0|δ). Since 1 + Mps(δ−β)p ≤2Mps(δ−β)p for s ∈ (0, 1], estimate (51.84) implies

tβ−δ|Φu0(u)(t)|β ≤12M + 2CACF MpB(α − β, 1− (β − δ)p)T α−δ−(β−δ)p < M

provided T is small enough. Hence Φu0 maps BM into BM for such T .Now let u, v ∈ BM . Then, similarly as above,

tβ−δ|Φu0(u)(t) − Φu0(v)(t)|β ≤ tβ−δ∣∣∣ ∫ t

0

e−(t−s)A(F (u(s))− F (v(s))

)ds∣∣∣β

≤ CACF tβ−δ

∫ t

0

(t− s)α−1−β |u(s)− v(s)|β(1 + |u(s)|p−1

β + |v(s)|p−1β

)ds

≤ 3CACF Mp−1B(α− β, 1− (β − δ)p)T α−δ−(β−δ)p‖u− v‖YT ,

hence‖Φu0(u)− Φu0(v)‖YT ≤

12‖u− v‖YT ,

provided T is small enough. Consequently, Φu0 is a contraction in BM and itpossesses a unique fixed point u.

Assume 0 < t1 < t2 ≤ T and let either t1 → t2− or t2 → t1+. Then

|u(t2)− u(t1)|γ ≤ ‖e−t1A‖L(Xδ,Xγ)|(e−(t2−t1)A − 1)u0|δ

+∫ t2

t1

‖e−(t2−s)A‖L(Xα−1,Xγ)|F (u(s))|α−1 ds

+∫ t1

0

‖e−(t1−s)A‖L(Xα−1,Xγ)|(e−(t2−t1)A − 1)F (u(s))|α−1 ds

→ 0,

due to (51.11), (51.83), u ∈ BM ,

e−(t2−t1)Au0 → u0 in Xδ,

e−(t2−t1)AF (u(s)) → F (u(s)) in Xα−1

and the Lebesgue theorem. Consequently, u ∈ C((0, T ], Xγ).The continuity of u : [0, T ]→ Xδ at t = 0 follows from the strong continuity of

the semigroup e−tA in Xδ and the estimate

|Φu0(u)(t)− e−tAu0|δ =∣∣∣ ∫ t

0

e−(t−s)AF (u(s)) ds∣∣∣δ

≤ CACF

∫ t

0

(t− s)−(δ+1−α)+(1 + |u(s)|pβ

)ds

≤ 2CACF MpB(min(1, α− δ), 1− (β − δ)p)tk,


where k = min(1, α− δ)− (β − δ)p.(ii) Let α = (β − δ)p + δ. First let us prove that

tβ−δ|e−tAu0|β → 0 as t→ 0. (51.85)

In fact, choose tk → 0 and set Sk := tβ−δk e−tkA. Then Sk ∈ L(Xδ, Xβ) are uni-

formly bounded due to (51.11) and, given w ∈ Xβ ,

|Skw|β ≤ tβ−δk |e−tkAw|β ≤ tβ−δ

k CA|w|β → 0 as k →∞.

Since Xβ is dense in Xδ, we obtain |Skw|β → 0 for any w ∈ Xδ. Consequently,(51.85) is true.

Choose M > 0 such that 2CACF Mp−1B(α−β, 1−(β−δ)p) ≤ 1/4 and T ∈ (0, 1]such that

tβ−δ|e−tAu0|β ≤ M/2 for all t ≤ T

(this choice is possible due to (51.85)). Let u ∈ BM = BM,T . Then

tβ−δ|Φu0(u)(t)|β ≤ tβ−δ|e−tAu0|β + CACF tβ−δ

∫ t

0

(t− s)α−1−β(1 + |u(s)|pβ

)ds

≤M/2 + CACFT α−δ

α− β+ CACF MpB(α − β, 1− (β − δ)p)

≤ 3M/4 + CACFT α−δ

α− β≤ M,

(51.86)provided T is small enough. Consequently, Φu0 maps BM into BM . Similar esti-mates (cf. (i)) show that Φu0 is a contraction and the corresponding fixed pointpossesses the required continuity properties.

Remarks 51.26. (i) It is easily seen from the proof that the existence time T inTheorem 51.25(ii) can be chosen uniform for initial data belonging to a compactsubset of Xδ.

(ii) Theorem 51.25 guarantees that the solution u is unique in the ball BM .However, similarly as in the proof of Theorem 15.2 one can prove the uniquenessof this solution in the space C := C([0, T ], Xδ) ∩ C((0, T ], Xβ). In fact, let v ∈ Cbe any solution of (51.9) on a (small) interval [0, τ ]. Then K :=v(t) : 0 ≤ t ≤ τis compact in Xδ hence (i) and the proof of Theorem 51.25 guarantee the solv-ability of (51.9) in BM,TK for some M, TK > 0 and for all initial data in K. LetU(t)v(s) denote the corresponding solution starting at v(s), s ∈ [0, τ ], t ∈ [0, TK ].Then tβ−δ|U(t)v(s)|β ≤ M . Fix s ∈ (0, min(TK , τ)) and denote u1(t) := U(t)v(s)and u2(t) := v(t + s). Then u1 ∈ C([0, TK ], Xβ) due to the existence part ofTheorem 51.7 and the uniqueness in Theorem 51.25, and u2 ∈ C([0, τ − s], Xβ).In addition, both u1 and u2 solve (51.9) with initial data v(s) ∈ Xβ . Hence

498 Appendices

u1 = u2 on [0, min(TK , τ − s)], due to the uniqueness in Theorem 51.7 (see alsoRemark 51.8(v)). Consequently,

tβ−δ|v(t + s)|β = tβ−δ|U(t)v(s)|β ≤ M.

Fix t > 0 small and let s → 0+ in the previous estimate. Then we obtaintβ−δ|v(t)|β ≤ M , hence v = u due to the uniqueness in Theorem 51.25.

The example in Remark 15.4(iii) and Example 51.27 below show that the re-striction v ∈ C((0, T ], Xβ) in the uniqueness statement above is necessary, ingeneral.

(iii) Let the assumptions of Theorem 51.25(i) be fulfilled. Similarly as in thecase of initial data in Xβ one can prove the existence of the maximal existence timeTmax = Tmax(u0), continuous dependence on initial data, positivity of the solutionu if Xδ is ordered and e−tA0 is positive, etc. For example, given u0, u0 ∈ Xδ, thereexists T = T (|u0|δ, |u0|δ) > 0 and C > 0 such that

|u(t)− u(t)|γ ≤ Ctδ−γ |u0 − u0|δ, t ≤ T, (51.87)

provided γ ∈ [δ, α).

(iv) A simple modification of the proof of Theorem 51.25 shows that the as-sumption β ≥ α− 1 is superfluous (cf. also Remark 51.8(v)).

(v) Assumption (51.82) in Theorem 51.25 can be replaced with

|F (u)− F (v)|α−1 ≤ CF

k∑i=1

|u− v|βi

(1 + |u|pi−1

βi+ |v|pi−1

βi

), (51.88)

where (for all i = 1, 2, . . . , k) pi > 1, 1 ≥ α > βi > δ ≥ −1, δ > βi − 1/pi,βi ≥ α − 1, α ≥ (βi − δ)pi + δ and F :

⋂ki=1 Xβi → Xα−1. In this case, it is

sufficient to use the fixed point argument in the space

YT :=

u ∈ L∞loc

((0, T ],

k⋂i=1

Xβi

): ‖u‖YT := max

isup

t∈(0,T )

tβi−δ|u(t)|βi < ∞.

For more complex generalizations of this type (and applications of such general-izations) see [446] and the references therein.

(vi) Estimate (51.84) implies

|u(t)|γ ≤ C(1 + t−(γ−δ)|u0|δ) for all γ ∈ [δ, α) and t ∈ (0, T ], (51.89)

where T is from Theorem 51.25.

Example 51.27. Let Ω, A and Xα, α ∈ [−1, 1], be as in Example 51.4(i), p > 1and q ≥ n(p − 1)/2, q > 1. Let F (u)(x) = f(x, u(x)), where f = f(x, u) is aC1-function satisfying f(·, 0) ∈ Lz(Ω) and the growth condition |∂uf(x, u)| ≤a(x) + C|u|p−1 with a ∈ Lp′z(Ω) and z ∈ (max(1, q/p), q], z < q if q = n(p− 1)/2(cf. Example 51.10). Assume u0 ∈ X0 = Lq(Ω) and set

β =12

(n

q− n

pz

), α =

12

(2 +

n

q− n

z

), δ = 0.

Then 1 ≥ α ≥ βp > 0 and α > βp if q > n(p − 1)/2, α < 1 if q = n(p − 1)/2.In addition, the choice of α and β guarantees Xβ → W 2β,q(Ω) → Lpz(Ω) andLz(Ω) → Xα−1 (since X1−α(q′) →W 2−2α,q′

(Ω) → Lz′(Ω)). Consequently, F sat-

isfies (51.82) and Theorem 51.25 guarantees the existence of a unique solution u ∈C([0, T ], Lq(Ω)) in the corresponding ball BM . In addition, u ∈ C((0, Tmax), Xγ)for any γ < α.

Let f = f(u) where f ′ is locally Holder continuous. If q > n(p− 1)/2, then wemay set z = q, hence α = 1 and

u ∈ C((0, Tmax), W 2γ,q ∩W 1,q0 (Ω)) for any γ < 1. (51.90)

If q = n(p − 1)/2, then we may choose z < q arbitrarily close to q, hence αarbitrarily close to 1, so that (51.90) remains true as well. Now Example 51.10guarantees

u ∈ C1((0, Tmax), W1,q0 (Ω))∩C((0, Tmax), W 2,q(Ω)) for any q ∈ [q,∞). (51.91)

In addition, Remark 51.26(iii) and (51.20) show

u(t; u0,k)→ u(t; u0) in BUC1(Ω) (51.92)

provided u0,k → u0 in Lq(Ω) and t ∈ (0, Tmax(u0)) is fixed.

Example 51.28. Let us consider the situation in Example 51.27, where Au =−∆u − λu, f(u) = |u|p−1, u0 ∈ Lq ∩ L2(Ω) with q ≥ qc = n(p− 1)/2, q > 1. Wewill show that the corresponding energy function

E(t) :=∫

Ω

(12|∇u(t)|2 − λ

2u2(t)− 1

p + 1|u(t)|p+1

)dx

is differentiable for t > 0. In addition, we will also prove that the problem generatesa dynamical system in H1

0 ∩ Lq(Ω) provided q ≥ max(qc, p + 1).Example 51.27 and Theorem 51.25 guarantee the existence of a unique maximal

solution u ∈ C([0, Tmax)), Lq(Ω)). In what follows we set T := Tmax. Let us firstshow that

u ∈ C([0, τ ], L2(Ω)) for some τ ∈ (0, T ). (51.93)

500 Appendices

If 2 ≥ n(p − 1)/2, then this assertion follows from the well-posedness in L2(Ω)(see Example 51.27). Hence assume 2 < n(p− 1)/2. Then q > 2. Let i ≥ 0 be theinteger such that 2pi < q ≤ 2pi+1.

First assume i = 0. Observe that estimate (15.2) in Theorem 15.2 and Remark15.4(i) remain obviously true for λ = 0. Applying (15.2) with r = 2p, we obtain

‖up(s)‖2 = ‖u(s)‖p2p ≤ C‖u(s)‖p

qs−θ ≤ Cs−θ, 0 < s < τ,

for some τ > 0, where up denotes |u|p−1u and

0 ≤ θ :=np

2

(1q− 1

2p

)=

n

2q(p− (q/2)) ≤ p− (q/2)

p− 1< 1.

Therefore up ∈ L1((0, τ), L2(Ω)), hence g(t) :=∫ t

0 e−(t−s)Aup(s) ds ∈ C([0, τ ],L2(Ω)), and (51.93) is satisfied.

Next assume i ≥ 1. For any r ∈ [2p, qp], we have

u ∈ C([0, τ ], Lr(Ω)) ⇒ up ∈ C([0, τ ], Lr/p(Ω)) ⇒ g ∈ C([0, τ ], Lr/p(Ω))

⇒ u ∈ C([0, τ ], Lr/p(Ω)),(51.94)

due to e−tAu0 ∈ C([0, τ ], L2∩Lq(Ω)). First applying (51.94) with r = q, we obtainu ∈ C([0, τ ], Lq/p(Ω)), hence u ∈ C([0, τ ], L2pi

(Ω)), due to q/p ≤ 2pi ≤ q. Thenapplying (51.94) iteratively with r = 2pi, 2pi−1, . . . , 2p, we end up with (51.93).

Due to (51.91) we know that there exists a positive constant C∞ such that|u| ≤ C∞ on Ω × [τ/2, T ]. Fix f ∈ BC1(R) such that f(u) = f(u) for |u| ≤ C∞.Then u is a solution of the equation ut − ∆u − λu = f(u) for t ≥ τ/2, henceestimate (51.91) (obtained with q = 2 and initial data u(τ/2)) implies

u ∈ C1([τ, T ), H10 (Ω)) ∩C([τ, T ), H2(Ω)).

Consequently, u ∈ C([0, T ), L2(Ω))∩C1((0, T ), H10 (Ω))∩C((0, T ), H2(Ω)). In par-

ticular, u ∈ C1((0, T ), L2(Ω)). Since also u ∈ C1((0, T ), Lq(Ω)) for any q ≥ q dueto (51.91), we have u ∈ C1((0, T ), Lp+1(Ω)), hence E ∈ C1((0, T )).

Next assume u0 ∈ H10 ∩ Lq∗(Ω), q∗ := max(qc, p + 1). We already know that

the solution satisfies u ∈ C([0, T ), L2 ∩ Lq(Ω)) for any q ∈ [qc,∞), q > 1, inparticular u ∈ C([0, T ), Lq∗

(Ω)). Let us prove u ∈ C([0, T ), H10 (Ω)). Since u ∈

C((0, T ), H10 (Ω)) and e−tAu0 ∈ C([0, T ), H1

0 (Ω)) it is sufficient to show∥∥∥ ∫ t

0

e−(t−s)A|u(s)|p−1u(s) ds∥∥∥

1,2→ 0 as t→ 0. (51.95)

Let Xθ(2), θ ∈ [0, 1], be the scale of spaces from Example 51.4(i) (in particular,X0(2) = L2(Ω) and X1/2(2) = H1

0 (Ω)). If qc = p + 1, then there exists ε > 0 suchthat Lq∗/p(Ω) → X−1/2+ε(2), hence (51.95) follows from

‖e−(t−s)A‖L(X−1/2+ε(2),X1/2(2)) ≤ (t− s)−1+ε

and ‖|u(s)|p−1u(s)‖q∗/p = ‖u‖pq∗ ≤ C. Let qc = p+1. Then Lq∗/p(Ω) → X−1/2(2).

Since the estimate (51.85) is uniform for u0 lying in a compact set of Xδ and theset |u(s)|p−1u(s) : s ∈ [0, T ] is compact in Lq∗/p(Ω), we have

‖e−(t−s)A|u(s)|p−1u(s)‖1,2 = o((t− s)−1

)as t→ 0. (51.96)

Now the smoothing estimate (15.2) with q = p + 1 = qc, r = 2p guarantees

‖|u(s)|p−1u(s)‖2 = ‖u(s)‖p2p ≤ C‖u(s)‖p

p+1s−1/2 ≤ Cs−1/2,

hence‖e−(t−s)A|u(s)|p−1u(s)‖1,2 ≤ C(t− s)−1/2s−1/2. (51.97)

Interpolation between (51.96) and (51.97) yields

‖e−(t−s)A|u(s)|p−1u(s)‖1,2 = o((t− s)−3/4s−1/4

)which guarantees (51.95). Consequently, u ∈ C([0, T ), H1

0 ∩ Lq∗(Ω)) and E ∈

C([0, T )). Similar estimates as above show the continuous dependence of solutionson initial data in H1

0 ∩ Lq∗(Ω), hence the problem generates a dynamical system

in this space. Obviously, the same remains true for the space H10 ∩ Lq(Ω) with

q ∈ (q∗,∞).If λ = 0, then the continuity properties of E can in some cases be proved without

the assumption u0 ∈ L2(Ω). For example, let 1 < p ≤ pS , Ω = Rn, u0 ∈ Lp+1(Rn)

and ∇u0 ∈ L2(Rn). Set q := p + 1 ≥ n(p− 1)/2. Then (51.91) shows

u ∈ C([0, T ), Lp+1(Rn)) ∩C1((0, T ), Lp+1(Rn)) ∩C((0, T ), W 2,q(Rn))

for any q ≥ p + 1. In addition, estimate (51.84) implies |u(t)|γ ≤ C(‖u0‖p+1)t−γ

for any γ < 1. If p = pS set γ = 1/(2p). Otherwise fix γ < 1/(2p) such thatXγ = W 2γ,p+1(Rn) → L2p(Rn) and set

v(t) :=∫ t

0

e−(t−s)AF (u(s)) ds.

Then

‖v(t)‖1,2 ≤∫ t

0

(t− s)−1/2‖F (u(s))‖2 ds ≤∫ t

0

(t− s)−1/2‖u(s)‖p2p ds

≤∫ t

0

(t− s)−1/2|u(s)|pγ ds ≤M

∫ t

0

(t− s)−1/2s−γp ds <∞

and‖v(t)‖1,2 → 0 as t→ 0, (51.98)

502 Appendices

due to M = M(t)→ 0 as t→ 0 if p = pS (cf. estimates in (51.86)). Since also

‖∇e−tAu0‖2 = ‖e−tA∇u0‖2 ≤ ‖∇u0‖2,

we obtain ∇u(t) ∈ L2(Rn). Similar estimates show the local Holder continuityof v : (0, T ) → H1(Rn) and ∇(e−tAu0) ∈ C1((0, T ), L2(Rn)). Since u : (0, T ) →BUC∩L2p(Rn) is locally Holder continuous due to interpolation and∇u : (0, T )→L2(Rn) is also locally Holder continuous, we have F (u) ∈ Cρ((0, T ), H1(Rn)) forsome ρ > 0. Finally, (51.98) and Theorem 51.1(v) imply v ∈ C1((0, T ), H1(Rn))∩C([0, T ), H1(Rn)). Since also ∇(e−tAu0) = e−tA(∇u0) ∈ C([0, T ), L2(Rn)) we seethat the energy function E belongs to C1((0, T )) ∩ C([0, T )).

Example 51.29. Let Ω, A and Xα, α ∈ [−1, 1], be as in Example 51.4(i), F (u) =|u|r−1u − µ|∇u|p, where µ ∈ R, p, r > 1, q > n(p − 1), 1/r > 1/p− q/n. Assumeu0 ∈ X1/2 = W 1,q

0 (Ω), choose z ∈ (max(1, q/p), q) such that 1/r ≥ 1/p− z/n andset

β =12

(1 +

n

q− n

pz

), α =

12

(2 +

n

q− n

z

), δ =

12.

Then 1 > α > (β − δ)p + δ > 0 and δ ∈ (β − 1/p, β). Since F2(u) := |∇u|p andF1(u) := |u|r−1u can be viewed as

F2 : Xβ →W 2β,q(Ω) →W 1,pz(Ω) ∇→ (Lpz(Ω))n |·|p−−→ Lz(Ω) → Xα−1,

F1 : Xβ →W 2β,q(Ω) → Lrz(Ω)|u|r−1u−−−−−→ Lz(Ω) → Xα−1,

we see that F satisfies (51.82). Now Theorem 51.26 and the same bootstrap argu-ment as in Example 51.27 guarantee the existence of T := Tmax and of a solutionu ∈ C([0, T ), W 1,q

0 (Ω))∩C((0, T ), W 2γ,q(Ω)), where γ < 1 and q ∈ [q,∞) are arbi-trary. Choose γ, q such that (2γ − 1)q > n. Then Wq := W 2γ−1,q(Ω) → BUC(Ω)and |∇u| ∈ C((0, T ), Wq ∩Wq). If w ∈Wq ∩Wq, w ≥ 0, then w ≤ C in Ω, hence

|wp(x)− wp(y)| ≤ pCp−1|w(x) − w(y)| (51.99)

and using the intrinsic norm in Wq (see [13], for example) we obtain wp ∈ Wq,‖wp‖Wq ≤ pCp−1‖w‖Wq . Since |∇u| ∈ C((0, T ), Wq ∩Wq) and Wq → BUC(Ω),Wq → Lq(Ω), using (51.99) we obtain |∇u|p ∈ C((0, T ), Lq(Ω)). This fact, the localboundedness of |∇u|p : (0, T ) → Wq and interpolation yield |∇u|p ∈ C((0, T ),W s,q(Ω)) for s ∈ (0, 2γ − 1), hence F2(u) ∈ C((0, T ), Xη) for η small enough.Similar estimates show F1(u) ∈ C((0, T ), Xη) for η small. Now Theorem 51.1(v)guarantees

u ∈ C1((0, T ), Lq(Ω)) ∩ C((0, T ), W 2,q ∩W 1,q0 (Ω)). (51.100)

Example 51.30. Let Ω, A and Xα, α ∈ [0, 1], be as in Remark 51.11, u0 ∈ X0

and F (u) = f(u,∇u), where f ∈ C1,

|fξ(u, ξ)| ≤M(|u|)(1 + |ξ|p−1), 1 ‖u0‖∞. Since Xβ → BC1(Ω), (51.82) is truewith CF = C(F, C∞) for all u, v ∈ Xβ satisfying ‖u‖∞, ‖v‖∞ ≤ C∞. Now anobvious modification of Theorem 51.25 shows the well-posedness of problem (51.4)in X0 ∈ L∞(Ω), BC(Ω) (see also [344, Theorem 7.1.6]).

Example 51.31. Let Ω, A and Xα, α ∈ [−1, 1], be as in Example 51.4(i), Ωbounded, p ∈ (1, 1 + 2/n), F (u) = ±|u|p−1u and u0 be a bounded Radon measurein Ω. Fix q ∈ (1, p) and choose δ such that

n− n

q< −2δ <

n + 2p

− n

q.

Notice that δ ∈ (−1, 0). Set α = 1 + δ and choose β such that

12

(n

q− n

p

)< β <

1p

+ δ.

Then u0 ∈ Xδ (since Xδ = (W−2δ,q′0 (Ω))′ and W−2δ,q′

0 (Ω) → C0(Ω)) and F :Xβ → Xα−1 (since Xβ → Lp(Ω) and L1(Ω) → Xδ = Xα−1). In addition, α >(β − δ)p + δ > 0 and δ ∈ (β− 1/p, β). Consequently, we can use Theorem 51.25 inorder to get a solution u ∈ C([0, T ], Xδ)∩C((0, T ], Xγ) for any γ < 1+δ. Choosingγ = 0 we obtain u ∈ C((0, T ], Lq(Ω)). Since q > 1 > n(p − 1)/2, Examples 51.27and 51.9 guarantee that u is a classical solution for t > 0.

Let us mention that the assumption p < 1 + 2/n is also necessary for thesolvability of (51.9) if u0 is the Dirac distribution, see [95].

Example 51.32. Consider the system

∂tu1 −∆u1 = |u2|p1−1u2, x ∈ Ω, t > 0,

∂tu2 −∆u2 = |u1|p2−1u1, x ∈ Ω, t > 0,

u1 = u2 = 0, x ∈ ∂Ω, t > 0,

u1(x, 0) = u0,1(x), u2(x, 0) = u0,2(x), x ∈ Ω,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(51.101)

where (u0,1, u0,2) ∈ X0 := Lr1 ×Lr2(Ω), p1, p2, r1, r2 > 1 and Ω ⊂ Rn is uniformly

regular of class C2. Assume

max( n

r2p1 −

n

r1,

n

r1p2 −

n

r2

)≤ 2. (51.102)

504 Appendices

Set X1 := W 2,r1 ∩W 1,r10 (Ω) ×W 2,r2 ∩W 1,r2

0 (Ω), Au := (−∆u1,−∆u2) for u =(u1, u2) ∈ X1, and let Xα be defined similarly as in Example 51.4(i). We will useRemark 51.26(v) in order to prove the well-posedness of (51.101) in X0.

Choose α ∈ (0, 1) with α > maxi(1 − n/(2r′i)), set δ = 0, βi = α/pi, i = 1, 2,and define zi ∈ (1, ri) by n/zi = n/ri + 2− 2α. Then Lz1 × Lz2(Ω) → Xα−1 and(51.102) guarantees W 2β1,r2(Ω) → Lp1z1(Ω), W 2β2,r1(Ω) → Lp2z2(Ω). Now it iseasy to verify (51.88), hence (51.101) is well-posed in X0. Theorem 32.1(ii) showsthat condition (51.102) is optimal.

Theorem 51.33. Let α, β, δ, p, F be as in Theorem 51.25. Assume, in addition,that ω(−A) < 0 and

|F (u)|α−1 = o(|u|β) as |u|β → 0

if α > (β − δ)p + δ,|F (u)|α−1 ≤ CF |u|pβ

if α = (β − δ)p + δ. Then, given ω ∈ (ω(−A), 0), there exists η > 0 and C > 0such that the solution u with initial data u0 satisfying |u0|δ < η exists globally and

|u(t)|β ≤ Ctδ−βeωt|u0|δ for all t ≥ 0. (51.103)

Proof. Fix η1 > 0 and assume |u0|δ ≤ η1. If α > (β − δ)p + δ, then estimate(51.87) with u0 = 0 shows that

|u(t)|β ≤ C1tδ−β |u0|δ for all t ∈ (0, T1], (51.104)

where T1 = T1(η1) ∈ (0, 1]. Let δ∗ > 0 be the constant from Theorem 51.17.Choose η > 0 such that C1T

δ−β1 η < δ∗. Then the conclusion follows from (51.104)

and Theorem 51.17 applied to the initial data u(T1).If α = (β−δ)p+δ choose η > 0 such that CF (C∗)pB(α−β, 1−(β−δ)p)ηp−1 < 1,

where C∗ := 2CA. Assume |u0|δ < η and set

T = supt ∈ (0, Tmax(u0)) : |u(s)|β ≤ C∗sδ−β |u0|δ for all s ∈ (0, t].Notice that T > 0, since u ∈ BM,T (M) and the constant M can be chosen arbi-trarily small in the proof of Theorem 51.25(ii). If T = ∞, then (51.104) is true fort ≤ T1 := 1 and we can proceed as in the case α > (β − δ)p + δ. Assume T < ∞.Then T < Tmax(u0), hence

|u(T )|β = C∗T δ−β|u0|δ. (51.105)

On the other hand,

|u(T )|β ≤ CAT δ−β|u0|δ + CACF

∫ T

0

(T − s)α−1−β |u(s)|pβ ds

≤ CAT δ−β|u0|δ + CACF (C∗)pB(α− β, 1− (β − δ)p)|u0|pδTδ−β

< C∗T δ−β|u0|δ,which yields a contradiction and concludes the proof.

51.6. Uniform bounds from Lq-estimates

In this part we present an abstract approach for obtaining L∞-bounds of solutionsfrom Lq-bounds. We will assume that (51.2) is true with ω < 0 and use the scale(Xα, Aα) introduced above.

The idea of the proof of the next proposition is contained in the proof of [14,Theorem 12.8].

Proposition 51.34. Let 0 ≤ β < α ≤ 1, −1 ≤ γ < β, T ∈ (0,∞] and Cγ > 0.Let F : Xβ → Xα−1 be continuous and

|F (u)|α−1 ≤ CF (|u|γ)(1 + |u|1−εβ ), u ∈ Xβ, (51.106)

where ε ∈ (0, 1). Let u0 ∈ Xβ and let u ∈ C([0, T ), Xβ) solve (51.9). If |u(t)|γ ≤ Cγ

for all t ∈ [0, T ), then |u(t)|β ≤ Cβ for all t ∈ [0, T ), where Cβ depends on Cγ and|u0|β but not on T .

Proof. Let T ∈ (0, T ) and t ≤ T . Using (51.3) we obtain

|u(t)|β ≤ |e−tAu0|β +∫ t

0

‖e−(t−s)A‖L(Xα−1,Xβ)|F (u(s))|α−1 ds

≤ c|u0|β + c

∫ t

0

eω(t−s)(t− s)α−1−βCF (Cγ)(1 + |u(s)|1−εβ ) ds

≤ c|u0|β + cCF (Cγ)∫ ∞

0

eωττα−1−β dτ(1 + sup

0≤s≤T

|u(s)|1−εβ

)and the assertion follows by choosing t such that |u(t)|β > sup0≤s≤T |u(s)|β − 1and letting T → T .

Remark 51.35. Let the hypothesis of Proposition 51.34 be satisfied with ε = 0.Then the proof and the singular Gronwall inequality in Proposition 51.6 guaranteethat |u(t)|β ≤ C1e

C2t.

Lemma 51.36. Let p > 1, −1 ≤ δ < (1 − 1/p)γ + β/p and

|F (u)|α−1 ≤ C(1 + |u|pδ

), u ∈ Xβ . (51.107)

Then the estimate (51.106) is true.

Proof. We can find θ ∈ (0, 1/p) such that (1− θ)γ + θβ > δ, hence (Xγ , Xβ)θ →Xδ and |u|δ ≤ |u|1−θ

γ |u|θβ. Now the assertion is obvious.

As an application we first give an alternative proof of Theorem 16.4. This proofwill not require Ω to be bounded.

506 Appendices

Proof of Theorem 16.4. Let Ω, A and Xα = Xα(q), α ∈ [−1, 1], be as inExample 51.4(i). Notice that we can choose ω < 0 if Au = −∆u + au, a > 0 (ora = 0 if Ω is bounded). Set F (u) = |u|p−1u + au, γ = 0 and α = 1. Using theassumption q > n(p− 1)/2 it is easy to find β < 1 close to 1 and δ < β/p close toβ/p such that Xδ → Lpq ∩ Lq(Ω). Consequently,

|F (u)|0 ≤ ‖u‖ppq + a‖u‖q ≤ C(1 + |u|pδ),

hence (51.107) is true. Now assuming ‖u(t)‖q ≤ C0, Lemma 51.36 and Proposi-tion 51.34 guarantee |u(t)|β < Cβ = Cβ(C0, |u0|β). Since Xβ → Lq(Ω) for someq > q, an obvious bootstrap argument shows ‖u(t)‖∞ < C∞ and concludes theproof.

Remarks 51.37. (i) If the assumptions of Theorem 16.4 are satisfied, then theabove proof guarantees the estimate U∞ ≤ C(u0)Uρ

q for suitable ρ ≥ 1.(ii) If we consider the more general problem

ut −∆u = f(x, t, u,∇u), x ∈ Ω, t > 0,

u = 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (51.108)

where f = f(x, t, u, ξ) is a C1-function satisfying the growth condition |f | ≤C(1 + |u|p + |ξ|r) with p < 1 + 2q/n and r < 1 + q/(n + q), then the Lq-bound forthe solution of (51.108) guarantees the L∞-bound. The same is true if one considers(51.108) with the nonlinear Neumann boundary condition ∂νu = g(x, t, u) insteadof the homogeneous Dirichlet condition, provided g ∈ C1 satisfies the growthcondition |g| ≤ C(1 + |u|z), z < 1 + q/n (see [14] and [435]).

(iii) Let 0 ≤ β < γ < α ≤ 1 and F : Xβ → Xα−1 be bounded on bounded sets.Let u0 ∈ Xβ and let u ∈ C([0,∞), Xβ) be a global solution of (51.9). If u(t) isuniformly bounded in Xβ for all t ≥ 0 and δ > 0, then the estimate

|u(t)|γ ≤ |e−tAu0|γ +∫ t

0

‖e−(t−s)A‖L(Xα−1,Xγ)|F (u(s))|α−1 ds

≤ Ctβ−γ |u0|β + C

∫ t

0

eω(t−s)(t− s)α−1−γ ds ≤ C(1 + tβ−γ),

implies the boundedness of u(t) in Xγ for t ≥ δ. If A has compact resolvent, thenthe embedding Xγ → Xβ is compact (see Theorem 51.1(i)), hence the trajectoryof u is relatively compact in Xβ.

Example 51.38. Let Ω ⊂ Rn be bounded with C2-boundary, −A be the Dirichlet

Laplacian and F (u) = f(u), where f ∈ C1, |f ′(u)| ≤ C(1 + |u|p−1), 1 < p < pS .Assume that u0 ∈W 1,2

0 (Ω) and u is a global solution of (51.9). If u(t) is uniformlybounded in Lp+1(Ω), then the trajectory of u is relatively compact in W 1,2

0 (Ω).

52. Appendix F: Maximum and comparison principles. Zero number 507

In fact, Example 51.10 shows the existence of T > 0 such that u ∈ C([0, T ],W 1,2

0 (Ω)) ∩ C((0, T ], W 2β,q(Ω)) for any β < 1 and q ≥ 1. Fixing q := p + 1 >n(p − 1)/2, η ∈ (0, T ), and considering the solution u on the interval [η,∞), theproof of Theorem 16.4 above shows that u(t) remains bounded in W 1,q

0 ∩W 2β,q(Ω)for some β close to 1. In particular, the solution remains bounded in W 1,2

0 (Ω) andthe assertion follows from Remark 51.37(iii).

Example 51.39. Let Ω ⊂ Rn be bounded with C2-boundary, −A be the Dirich-

let Laplacian and F : L∞(Ω) → L∞(Ω) be uniformly Lipschitz continuous onbounded subsets of L∞(Ω). Assume that u0 ∈ L∞(Ω) and u is a global solution of(51.9) which is uniformly bounded in L∞(Ω). Then the trajectory u(t) : t ≥ 1is relatively compact in L∞(Ω). This follows from Remark 51.11 and (the corre-sponding analogue of) Remark 51.37(iii).

52. Appendix F: Maximum and comparisonprinciples. Zero number

Maximum and comparison principles represent a very useful tool in the study ofscalar equations (and of some particular systems). Unfortunately, it is not easy toprovide (or find in the literature) a general statement which would be applicablein all situations. We therefore prove — or at least formulate — various versionsof these principles which we frequently use. For simplicity we have stated all theresults for the case when the elliptic part of the equation is the Laplacian, butthey remain true for more general operators (under suitable assumptions).

52.1. Maximum principles for the Laplace equation

We first recall the weak and strong maximum principles and the Hopf bound-ary lemma for strong subsolutions (cf. [250, Theorems 9.1, 9.6, and the proof ofLemma 3.4]).

Proposition 52.1. Let Ω be an arbitrary domain in Rn, b ∈ L∞(Ω, Rn), and let

u ∈W 2,nloc (Ω) satisfy

−∆u + b · ∇u ≤ 0 a.e. in Ω. (52.1)

(i) If u ∈ C(Ω), u ≤ 0 on ∂Ω, and Ω is bounded, then u ≤ 0 in Ω.(ii) If u ≤ 0 in Ω, then either u ≡ 0 or u < 0 in Ω.(iii) Let x0 ∈ ∂Ω. Assume that Ω satisfies an interior sphere condition at x0 andthat u is continuous at x0. If u ≤ 0 in Ω and u(x0) = 0, then

lim inft→0+

t−1u(x0 − tν) < 0.

508 Appendices

(Here the outer normal ν is defined in the natural way via the interior sphere atx0). In particular, we have

∂νu(x0) > 0

whenever this derivative exists.

Remark 52.2. Assertions (ii) and (iii) of Proposition 52.1 remain valid if theinequality in (52.1) is replaced with −∆u+ b ·∇u+ cu≤ 0 for some constant c > 0(cf. e.g. [239]). This follows easily by applying Proposition 52.1(ii) and (iii) to thefunction v(x) = eαx1u(x) with α > 0 large enough.

We next give a useful maximum principle under weaker regularity assumptions,namely for variational or, even, distributional subsolutions.

Proposition 52.3. Let Ω be an arbitrary domain in Rn and let u ∈ L1

loc(Ω)satisfy

−∆u ≤ 0 in D′(Ω).

Assume that either:(i) u ∈ H1(Ω) and u ≤ 0 on ∂Ω in the sense that u+ ∈ H1

0 (Ω); or(ii) Ω is bounded, u is continuous in a neighborhood of ∂Ω and u ≤ 0 on ∂Ω.

Then u ≤ 0 a.e. in Ω.

Proof. We first assume (i). We shall use the Stampacchia truncation argument.By assumption we have∫

Ω

∇u · ∇ϕdx ≤ 0, for all 0 ≤ ϕ ∈ D(Ω). (52.2)

Fix a C∞-function G : R → R+ such that G(s) = 0 for s ≤ 0 and 0 < G′(s) ≤ 1for s > 0. By our assumption that u+ ∈ H1

0 (Ω), there exists a sequence ψj ∈ D(Ω)such that ψj → u+ in H1(Ω) and a.e. Let ϕj = G ψj . We have 0 ≤ ϕj ∈ D(Ω).Writing

|∇(G ψj)−∇(G u+)| ≤ G′(ψj)|∇ψj −∇u+|+ |G′(ψj)−G′(u+)||∇u+|,

we obtain ∇ϕj → ∇(G u+) in L2(Ω) by dominated convergence. Since ∇u+ =χu>0∇u, it follows from (52.2) that∫

Ω

G′(u+)|∇u+|2 dx =∫

Ω

∇u · ∇(G u+) dx = limj→∞

∫Ω

∇u · ∇ϕj dx ≤ 0.

Consequently, ∇(u+)2 = 2u+∇u+ = 0 a.e. in Ω. Since u+ ∈ H10 (Ω), we conclude

that u+ = 0 a.e. in Ω.Let us next consider case (ii). For ε > 0, denote ωε = x ∈ Ω : δ(x) > ε. By

assumption, there exists ε0 > 0 small, such that u is continuous on Ω \ ωε0 . Now

set uj := ρj ∗ u, where ρj is a sequence of mollifiers defined by (47.6), and fixε ∈ (0, ε0). For j ≥ j0(ε) large, we have uj ∈ C2(ωε) and ∆uj = ∆u ∗ ρj ≥ 0 inωε. Therefore, the assertion in case (i) implies supωε

uj ≤ sup∂ωεuj. Since uj → u

in L1(ωε) and in C(∂ωε), it follows that ess supωεu ≤ sup∂ωε

u. The conclusionfollows by letting ε → 0 and using the fact that limε→0(sup∂ωε

u) ≤ 0.

In the rest of Appendix F we shall only consider parabolic problems.

52.2. Comparison principles for classical and strongsolutions

We start with a basic maximum principle for classical solutions. Note that un-bounded and singular first-order coefficients are allowed (this will be used inLemma 52.18 below).

Proposition 52.4. Let Ω be an arbitrary domain in Rn, T > 0, b : QT → R

n,c : QT → R, with supQT

c < ∞. Assume that w = w(x, t) ∈ C2,1(QT ) ∩ C(QT )satisfies w ≤ 0 on PT , supQT

w <∞, and

wt −∆w ≤ b · ∇w + cw in QT . (52.3)

If Ω is unbounded, assume in addition that either

lim sup|x|→∞, (x,t)∈QT

w(x, t) ≤ 0, (52.4)

or|b(x, t)| ≤ C1(1 + |x− a|−1), x ∈ QT , (52.5)

for some a ∈ Rn and C1 > 0. Then w ≤ 0 in QT .

Proof. We may assume c < 0 (if this is not true, then it is sufficient to considerthe function w(x, t) = e−λtw(x, t), where λ > supQT

c). Also, we may obviouslyassume w ∈ C2,1(Ω× (0, T ]).

Case 1: Ω bounded. Assume on the contrary that w achieves a positive interiormaximum at some point (x0, t0) ∈ Ω×(0, T ]. At this point we have w > 0,∇w = 0,∆w ≤ 0, wt ≥ 0. Using c < 0 we obtain

0 ≤ wt −∆w − b · ∇w ≤ cw < 0,

which yields a contradiction.Case 2: Ω unbounded. If the conclusion is not true, then we have w(x0, t0) > 0

for some (x0, t0) ∈ QT . In case (52.4) is satisfied, then w achieves its positivemaximum and we conclude as in case 1. In case (52.5) holds, arguing similarly asin [296], we set

v(x, t) = w(x, t) − δt− ε(1 + |x− a|2)1/2,

510 Appendices

where δ, ε > 0 are such that v(x0, t0) > 0 and δ > ε(n + 2C1). We compute

∇(1 + |x− a|2)1/2 = (x− a)(1 + |x− a|2)−1/2,

∆(1 + |x− a|2)1/2 = (n + (n− 1)|x− a|2)(1 + |x− a|2)−3/2 ≤ n.(52.6)

Since v ≤ 0 on St0 , v attains its (positive) maximum in Qt0 at some (x1, t1) ∈Ω× (0, t0]. At this point we have w > v > 0, ∇v = 0, ∆v ≤ 0, vt ≥ 0. Using c ≤ 0,it follows that

0 ≤ vt = wt − δ ≤ ∆w + b · ∇w + cw − δ

≤ ∆v + b · ∇v + nε + ε|b||x− a|(1 + |x− a|2)−1/2 − δ

≤ ε(n + 2C1)− δ < 0,

which yields a contradiction and concludes the proof.

Remark 52.5. The assumption supQTc < ∞ in Proposition 52.4 is necessary

(although it can be sometimes weakened). Consider for instance the simple exam-ples u(x, t) = tϕ1(x), c(x, t) = λ1 + t−1 (Ω bounded), or u(x, t) = t, c(x, t) = t−1

(Ω = Rn), which satisfy ut −∆u = cu and u > 0 in QT , with u ≡ 0 on PT .

We next give a version of the comparison principle for classical (sub-/super-)solutions.

Proposition 52.6. Let Ω be an arbitrary domain in Rn, T > 0, u, v ∈ C2,1(QT )∩

C(QT ). Assume that u ≤ v on PT and

∂tu−∆u− f(x, u,∇u) ≤ ∂tv −∆v − f(x, v,∇v) in QT , (52.7)

where f = f(x, s, ξ) : Ω × R × Rn → R is continuous in x and C1 in s and ξ.

Assume also that

u, v,∇v ∈ L∞(QT ), |u|, |v| ≤ C1, |∇v| ≤ C2 (52.8)

and

|fs(x, s, ξ)|+(1+ |x|)−1|fξ(x, s, ξ)| ≤ Cf for all |s| ≤ C1, |ξ| ≤ C2 +1. (52.9)

Then u ≤ v in QT .

Proof. Fix τ ∈ (0, T ) such that τeCf τ < 1/8Cf . It is sufficient to prove u ≤ v inQτ . Assume on the contrary δ := supQτ

(u− v) > 0 and choose (x0, t0) ∈ Qτ suchthat (u− v)(x0, t0) > δ/2. Consider ε ∈ (0, 1) such that

ε < min(Cf δ/(n + Cf ), e−Cfτ , e−Cf t0δ/4ψ(x0))

and setz(x, t) = e−Cf t(u − v)(x, t)− 2Cfδt− εψ(x),

where ψ(x) = (1+ |x|2)1/2. Then z(x0, t0) > 0 and z attains its maximum in Qτ atsome (x, t) ∈ Qτ , since z(x, t)→ −∞ as |x| → ∞, uniformly in t. Now z(x, t) > 0implies (x, t) ∈ Qτ \Pτ , hence zt−∆z ≥ 0 and ∇z = 0 at this point. Consequently,

|∇u(x, t)−∇v(x, t)| ≤ eCf τε|∇ψ(x)| < 1,

since |∇ψ| ≤ 1. In addition, z(x, t) > 0 implies

ε|x| ≤ εψ(x) < e−Cf tδ ≤ δ.

Now the mean value theorem guarantees the existence of s between u(x, t), v(x, t)and ξ between ∇u(x, t), ∇v(x, t) such that

0 ≤ (zt −∆z)(x, t)

≤ e−Cf t[f(x, u(x, t),∇u(x, t))− f(x, v(x, t),∇v(x, t))

]− Cf (z(x, t) + 2Cfδt + εψ(x))− 2Cfδ + εn

= e−Cf t[fs(x, s,∇v(x, t))eCf t(z(x, t) + 2Cfδt + εψ(x))

+ fξ(x, u(x, t), ξ)eCf t(∇z(x, t) + ε∇ψ(x))]

− Cf (z(x, t) + 2Cfδt + εψ(x))− 2Cfδ + εn

≤ Cf (1 + |x|)ε− 2Cfδ + εn

< −Cfδ + ε(n + Cf ) < 0,

which yields a contradiction and concludes the proof.

The following proposition is a version of the strong comparison principle andof the Hopf boundary lemma (for strong solutions, in bounded domains). A moregeneral version can be derived by using the maximum principles in [152] (cf. theproof).

Proposition 52.7. Let Ω be a bounded domain in Rn of class C2, p > n+2, and

T > 0. Let u, v ∈W 2,1;ploc (Ω× (0, T ]) ∩ C([0, T ], L2(Ω)) ∩ L∞(QT ). Assume

∂tu−∆u− f(x, t, u,∇u) ≤ ∂tv −∆v − f(x, t, v,∇v) in QT ,

where f = f(x, t, s, ξ) : Ω× [0, T ]×R×Rn → R is continuous in x, t and C1 in s

and ξ. Assume also that u(·, 0) ≤ v(·, 0), u(·, 0) ≡ v(·, 0), and either

u ≤ v on ST (52.10)

512 Appendices

or∂νu + bu ≤ ∂νv + bv on ST , (52.11)

where b ∈ C1(∂Ω). Finally, if f depends on ξ, we also assume that ∇u,∇v ∈L∞(QT ). Then

u < v in QT .

In addition, if u(x0, t0) = v(x0, t0) for some x0 ∈ ∂Ω and t0 ∈ (0, T ), then

∂νu(x0, t0) > ∂νv(x0, t0).

If (52.11) is true, then u < v in Ω× (0, T ).

Proof. Setting w := v − u, the mean value theorem implies

∂wt −∆w ≥ g1(x, t)w + g2(x, t) · ∇w,

where g1(x, t) =∫ 1

0fu(x, t, u + θ(v − u),∇v) dθ and g2(x, t) =

∫ 1

0fξ(x, t, u,∇u +

θ(∇v −∇u)) dθ.Let us first consider the case u, v ∈ W 2,1;p(QT ) (hence in particular u, v ∈

C1,0(QT )). Then the assertion follows from [152, Propositions 13.1, 13.2 and The-orem 13.5]. Note that the proofs in [152] use a result from [165] and the strongmaximum principle for classical solutions (cf. [429] and [214]).

In the general case, since g1, g2 ∈ L∞(QT ) due to our assumptions, we may firstapply Proposition 52.8 and Remark 52.9 below to deduce that u ≤ v in QT . Sinceu(·, t) ≡ v(·, t) for all sufficiently small t > 0 due to u, v ∈ C([0, T ], L2(Ω)), theconclusion follows from the previous case.

52.3. Comparison principles via the Stampacchia method

We now give versions of the weak maximum and comparison principles which applyto W 2,1;2

loc sub-/supersolutions and discontinuous initial data (as well as possiblyunbounded domains).

Proposition 52.8. Let 0 < T < ∞. Let Ω be an arbitrary domain in Rn, c be

measurable and a.e. finite on QT with supQTc < ∞, and K ≥ 0. Assume that

w ∈ C(Ω× (0, T )) ∩ C([0, T ), L2loc(Ω)) satisfies

supQT

w <∞, wt,∇w, D2w ∈ L2loc(QT ).

If w ≤ 0 on PT and

wt −∆w ≤ K|∇w|+ cw a.e. in QT ,


thenw ≤ 0 in QT .

Proof. Let ε > 0, λ = supQTc, and set

z = we−λt − εψ,

whereψ(x, t) = Mt + (1 + |x|2)1/2, (52.12)

with M = n + K. We see that, a.e. in QT , there holds

∂tz −∆z −K|∇z| ≤ e−λt(c− λ)w + ε(−ψt + ∆ψ + K|∇ψ|)≤ ε(−M + n + K) ≤ 0.

(52.13)

We next apply the Stampacchia truncation method. Note that, for R > 0 largeenough and for each τ > 0, there exists η = η(τ) > 0 such that

z ≤ 0 inx ∈ Ω : δ(x) ≤ η or |x| ≥ R

× (τ, T − τ).

Our assumptions thus imply z+ ∈ C([0, T ), L2(Ω))∩H1loc((0, T ), L2(Ω)), z+(0) = 0

and, for a.e. t ∈ (0, T ), z+(t) ∈ H10 (Ω ∩ BR). For a.e. t ∈ (0, T ), since ∆z(·, t) ∈

L2(Ω ∩BR), ∇(z+)(·, t) = χz>0∇z(·, t), it follows from (52.13) that

12

d

dt

∫Ω

(z+)2(t) dx ≤ −∫

Ω

|∇(z+)|2 dx + K

∫Ω

|∇z|z+ dx

≤ −∫

Ω

|∇(z+)|2 dx +∫

Ω

|∇(z+)|2 dx +K2

4

∫Ω

(z+)2 dx

=K2

4

∫Ω

(z+)2 dx.

By integration, we conclude that z+ = 0 in QT and the conclusion follows byletting ε → 0.

Remark 52.9. Proposition 52.8 can be extended to the case of Neumann bound-ary conditions. For instance, assume that Ω is smooth and bounded, and thatw satisfies the assumptions of Proposition 52.8 with w ≤ 0 on PT replaced byw(·, 0) ≤ 0 and ∂νw + bw ≤ 0 on ST , where ∇w ∈ C(Ω× (0, T )) and b ∈ L∞(∂Ω).Then we conclude that w ≤ 0 in QT . This follows from simple modifications of theabove proof, with ε = 0, using the trace inequality ‖v‖L2(∂Ω) ≤ η‖∇v‖2+C(η)‖v‖2,v ∈ H1(Ω), applied with η > 0 small and v = z+(t) for a.a. t.

514 Appendices

Proposition 52.10. Let 0 < T < ∞, Ω be an arbitrary domain in Rn, and let

f = f(s, ξ) : R× Rn → R, be a C1-function. Let u ∈ C(Ω× (0, T )) satisfy

u ∈ C([0, T ), L2loc(Ω)), u ∈ L∞(QT ), ut,∇u, D2u ∈ L2

loc(QT ),

and similarly for v. If f depends on ξ, we also assume that ∇u,∇v ∈ L∞(QT ). Ifu ≤ v on PT and

ut −∆u − f(u,∇u) ≤ vt −∆v − f(v,∇v) a.e. in QT ,

thenu ≤ v in QT .

Proof. Let w = u− v and set

M := max(ess sup

QT

(|u|+ |∇u|), ess supQT

(|v|+ |∇v|))< ∞ (52.14)

andK := sup|fs(s, ξ)|+ |fξ(s, ξ)| : |s|, |ξ| ≤ M < ∞.

Letting c(x, t) = (f(u,∇u) − f(v,∇u))/(u − v) (defined to be 0 whenever thedenominator vanishes), we have |c| ≤ K and

wt −∆w ≤ f(u,∇u)− f(v,∇v) = c(u− v) + (f(v,∇u)− f(v,∇v))

≤ cw + K|∇w|

a.e. in QT . The result then follows from Proposition 52.8 applied to w.

Remarks 52.11. (a) In Proposition 52.4 (resp., Proposition 52.6) it is sufficient toassume that (52.3) (resp., (52.7)) holds in the set QT := (x, t) ∈ QT : w(x, t) > 0(resp., QT := (x, t) ∈ QT : u(x, t) > v(x, t)). A similar remark holds for Propo-sitions 52.8 and 52.10. Moreover any boundedness assumption on the functionsu, v,∇u,∇v needs to be verified only on the set QT .

(b) The proof of Proposition 52.6 shows that we can assume ∇u ∈ L∞(Ω)instead of ∇v ∈ L∞(Ω). In addition, we do not need to assume the boundedness of∇v (or ∇u) at all if f is independent of ξ. Similarly, the assumption u, v ∈ L∞(Ω)can be replaced by supQT

(u− v) <∞ if f is independent of u.

(c) In Proposition 52.10, assume f(s, ξ) to be only continuous (instead of C1)at s = 0, and suppose in addition that infQT |u| > 0 or infQT |v| > 0. Then theconclusion remains valid. Indeed, assume for instance σ := infQT |v| > 0 and letK0 := sup|f(s, ξ)| : |s| ≤M, |ξ| ≤M and

K1 := sup|fs(s, ξ)|+ |fξ(s, ξ)| : σ/2 ≤ |s| ≤ M, |ξ| ≤M,

with M defined by (52.14). Then the function c in the proof verifies |c(x, t)| ≤ K1

if |u(x, t)| ≥ σ/2, and |c(x, t)| ≤ 4K0/σ if |u(x, t)| < σ/2. A similar remark holdsconcerning Proposition 52.22 (systems).

(d) The proof of Proposition 52.10 shows that it is sufficient to assume that uor v ∈ L∞(QT ), and that supQT

(u− v) <∞.(e) In Proposition 52.6, if f is of the form f = f(u) + g(x,∇u), then the

assumptions (52.8)–(52.9) can be replaced by

lim sup|x|→∞, (x,t)∈QT

(u− v)(x, t) ≤ 0

and u or v ∈ L∞(QT ). This can be proved easily by using Proposition 52.4 andRemark (a) above.

(f) In Propositions 52.4 and 52.8, if c ≤ 0 and if, instead of w ≤ 0 on PT , weassume w ≤ M on PT for some M > 0, then the conclusion is w ≤M in QT (justapply the result to the function w −M).

(g) When comparing a solution with a sub-/supersolution, the above (and sim-ilar) results are usually applied on the time interval (0, T ) for each T < Tmax(u0),hence guaranteeing the boundedness of the solution (and possibly of its deriva-tives).

52.4. Comparison principles via duality arguments

We now provide “very weak” versions of the maximum and comparison principles,which are useful in particular in the study of complete blow-up (see Section 27).They can be also applied to show monotonicity of solutions in time (cf. Proposi-tion 52.20 below).

Assume that Ω is a bounded domain of class C2+α for some α ∈ (0, 1). LetT > 0, u0 ∈ L1

δ(Ω) and f ∈ L1loc([0, T ), L1

δ(Ω)). We say that u ∈ L1loc(Ω× [0, T )) is

a very weak supersolution of

ut −∆u = f, x ∈ Ω, t ∈ (0, T ),

u = 0, x ∈ ∂Ω, t ∈ (0, T ),

u(x, 0) = u0(x), x ∈ Ω,

⎫⎪⎬⎪⎭ (52.15)

if ∫ τ

0

∫Ω

u(ϕt + ∆ϕ) + fϕ

dx ds +

∫Ω

u0ϕ(0) dx ≤ 0 (52.16)

for any 0 < τ < T and any 0 ≤ ϕ ∈ C2,1(Ω× [0, τ ]) such that ϕ = 0 on ∂Ω× [0, τ ]and ϕ(τ) = 0. Subsolutions are defined similarly (namely, u is a subsolution if−u is a supersolution). Of course, the definition immediately carries over to thenonlinear case f = f(u).

516 Appendices

Remarks 52.12. (i) If u ∈ C2,1(Ω × (0, T )) ∩ C([0, T ), L1(Ω)) is a classical su-persolution of (52.15) (i.e., satisfies (52.15) with = signs replaced by ≥), then it iseasy to show that it is a very weak supersolution.

(ii) Alternatively, one could replace the integrability assumption on f neart = 0 by a continuity assumption on u (namely, u ∈ C([0, T ), L1

δ(Ω)) and justf ∈ L1

loc((0, T ), L1δ(Ω)) and adopt a definition more similar to that of weak L1

δ-solution (cf. Definition 48.8). However, the present formulation seems better suitedto certain applications, such as complete blow-up.

Proposition 52.13. Let Ω be a bounded domain of class C2+α for some α ∈(0, 1). Let 0 < T < ∞ and c ∈ L∞(QT ).(i) Assume that

z ∈ Lqloc(Ω× [0, T )) for some 1 < q < ∞. (52.17)

If z is a very weak supersolution ofzt −∆z = cz, x ∈ Ω, 0 < t < T,

z = 0, x ∈ ∂Ω, 0 < t < T,

z(x, 0) = 0, x ∈ Ω,

⎫⎪⎬⎪⎭ (52.18)

then z ≥ 0 a.e. in QT .(ii) If c = 0, then assertion (i) remains true for q = 1.

Proof. (i) Fix m > max(n/2, q′) and a sequence of functions cj ∈ D(QT ) suchthat cj → c in Lm(QT ). For given 0 < τ < T and 0 ≤ h ∈ D(Qτ ), let ϕj ∈ C2,1(Qτ )be the solution of

−∂tϕj −∆ϕj = cjϕj + h, x ∈ Ω, 0 < t < τ,

ϕj = 0, x ∈ ∂Ω, 0 < t < τ,

ϕj(x, τ) = 0, x ∈ Ω.

⎫⎪⎬⎪⎭ (52.19)

By Proposition 52.8, we have ϕj ≥ 0. Moreover, by using the variation-of-constantsformula, the Lm-L∞-estimate (Proposition 48.4), and m > n/2, one easily gets

‖ϕj‖L∞(Qτ ) ≤ C, j = 1, 2, . . . . (52.20)Applying the definition of z being a (very weak) supersolution of (52.18), withϕ = ϕj as a test-function, we obtain

0 ≤ −∫ τ

0

∫Ω

z(∂tϕj +∆ϕj + cϕj) dx ds =∫ τ

0

∫Ω

(hz+(cj− c)zϕj

)dx ds. (52.21)

Since (cj − c)zϕj → 0 in L1(Qτ ) due to (52.17), (52.20) and m > q′, we deducethat

∫ τ

0

∫Ω

hz dx ds ≥ 0, and the conclusion follows.(ii) The argument is much simpler than in the previous case: It suffices to use

(52.21) with c = cj = 0 and ϕ instead of ϕj , where ϕ is the solution of (52.19)with cj = 0.

We have the following (very weak) comparison principle for the semilinear prob-lem (14.1).

Proposition 52.14. Let Ω be a bounded domain of class C2+α for some α ∈(0, 1), 0 < T < ∞, and u0 ∈ L∞(Ω). Assume that f : R → R is of class C1.Let u, v ∈ L∞(QT ) be, respectively, very weak sub- and supersolutions to problem(14.1) on (0, T ). Then u ≤ v on (0, T ).

Proof. This is an immediate consequence of Proposition 52.13 applied to z :=v − u.

The comparison results in the previous subsections do not apply in the case ofconvective equations like

ut −∆u = |u|p−1u + a · ∇(|u|q−1u)

with 1 < q < 2, due to the fact that the nonlinearity is not C1 at u = 0. For suchproblems, we shall rely instead on the following result, the proof of which involvesa duality argument. For simplicity we restrict ourselves to the case Ω bounded orΩ = R

n.

Proposition 52.15. Let Ω be a bounded domain of class C2 or Ω = Rn. Let

T > 0, b ∈ L∞(QT , Rn) and c ∈ L∞(QT ). Assume that w = w(x, t) ∈C2,1(Ω× (0, T )) ∩ L∞(QT ) and that bw ∈ C1,0(Ω× (0, T )). If Ω = R

n assume inaddition that ∇w ∈ L∞

loc((0, T ), L∞(Rn)). If w ≤ 0 on ST , lim supt→0 w(x, t) ≤ 0for all x ∈ Ω, and

wt −∆w ≤ div(bw) + cw in QT , (52.22)

then w ≤ 0 in QT .

Proof. Fix h ∈ D(Ω), h ≥ 0, 0 < t2 < T .First consider the case Ω bounded and let ϕ be the solution of the adjoint

problem

−ϕt −∆ϕ = −b · ∇ϕ + cϕ, x ∈ Ω, 0 < t < t2,

ϕ = 0, x ∈ ∂Ω, 0 < t < t2,

ϕ(x, t2) = h(x), x ∈ Ω.

⎫⎪⎬⎪⎭ (52.23)

By parabolic Lr-regularity, we have ϕ ∈ W 2,1:r(QT ), 1 < r < ∞, and ϕ ≥ 0 byProposition 52.8. For each 0 < t1 < t2, multiplying (52.22) by ϕ, integrating byparts and using w ≤ 0 on PT , ∂ϕ/∂ν ≤ 0 = ϕ on ∂Ω, we obtain[∫

Ω

wϕdx

]t2

t1

=∫ t2

t1

∫Ω

(wϕt + wtϕ

)dx ds

≤∫ t2

t1

∫Ω

(wϕt + (∆w + div(bw) + cw)ϕ

)dx ds

≤∫ t2

t1

∫Ω

(ϕt + ∆ϕ− b · ∇ϕ + cϕ

)w dxds = 0.

(52.24)

518 Appendices

Letting t1 → 0, we obtain∫Ω

w(t2)h dx ≤ 0, hence w ≤ 0.

Next consider the case Ω = Rn. Observe that problem (52.23) still admits a

solution ϕ ∈ C([0, T ], W 1,1(Rn)), ϕ ≥ 0. This follows from a straightforward fixed-point argument, using the variation-of-constants formula and simple estimatesinvolving the Gaussian heat kernel G. Moreover, given 1 < r < ∞, we haveϕ ∈ C([0, T ], W 1,r(Rn)) due to Appendix E and ϕ ∈W 2,1:r

loc (QT ) by Theorem 48.1,and a simple cut-off argument. For R > 0, arguing as in (52.24) with Ω replacedby BR, we get[∫

BR

wϕdx

]t2

t1

≤∫ t2

t1

∫∂BR

(ϕ

∂w

∂ν− w

∂ϕ

∂ν+ (b · ν)wϕ

)dσds

≤ C(t1)∫ t2

t1

∫∂BR

(ϕ + |∇ϕ|

)dσds.

(52.25)

Since ϕ ∈ C([0, T ], W 1,1(Rn)) there exists a sequence Rj →∞ such that the RHSof (52.25) with R = Rj decays to 0. Then letting t1 → 0, we obtain

∫Rn w(t2)h dx =

0, hence w ≤ 0.

As a direct consequence of Proposition 52.15, we obtain in particular:

Proposition 52.16. Let Ω be a bounded domain of class C2 or Ω = Rn. Let

T > 0 and f, g : (t, u) [0, T ]×R→ R be such that f, fu, g, gu are continuous. Letu, v ∈ C2,1(Ω × (0, T )) ∩ L∞(QT ). If Ω = R

n assume in addition that ∇u,∇v ∈L∞

loc((0, T ), L∞(Rn)). If u ≤ v on ST , lim supt→0(u − v)(x, t) ≤ 0 for all x ∈ Ω,and

∂tu−∆u− f(t, u)− div(g(t, u)) ≤ ∂tv −∆v − f(t, v)− div(g(t, v)) in QT ,

then u ≤ v in QT .

52.5. Monotonicity of radial solutions

Assume that Ω is a symmetric domain and that problem (34.1) is well-posed in aspace of functions X on Ω. If the C1-function F = F (s, ξ) depends on ξ through|ξ| only and if u0 ∈ X is radial, then the solution u of (34.1) is also radial. Thisfollows immediately from the local uniqueness and the invariance of problem (34.1)by rotation. The same remains true in the case of Neumann boundary conditions.The following result provides sufficient conditions for the preservation of radialmonotonicity.

Proposition 52.17. Let Ω = BR or Ω = Rn. In what follows we use the notation

T = Tmax(u0).

(i) Consider problem (34.1) with a C1-function F = F (s, ξ) : R+ × Rn → R such

that F (s, ξ) = F (s, |ξ|) and F (0, 0) ≥ 0. Assume that u0 ∈ BC1(Ω), u0 = 0 on∂Ω, u0 ≥ 0, is radial nonincreasing. Then

u ≥ 0 and u is radial nonincreasing in QT . (52.26)

(ii) Consider problem (14.1) with f ∈ C1([0,∞)) such that f(0) ≥ 0. If u0 ∈L∞(Ω), u0 ≥ 0, is radial nonincreasing, then (52.26) is true.(iii) Consider problem (15.1) with p > 1, and let 1 ≤ q < ∞ satisfy q > qc =n(p − 1)/2 or q = qc > 1. If u0 ∈ Lq(Ω), u0 ≥ 0, is radial nonincreasing, then(52.26) is true.

Moreover, in each case above, if u0 ≡ 0, then

ur < 0 in (0, R]× (0, T ) (52.27)

(with (0, R] replaced by (0,∞) if Ω = Rn).

We need the following lemma:

Lemma 52.18. Let Ω = (0, R), 0 < R ≤ ∞, T > 0, f = f(s, ξ) ∈ C1(R+ × R).Assume that u ∈ C2,1(Ω× (0, T ))∩BC(QT ) satisfies ur ∈ W 2,1;2

loc (QT )∩BC(QT ),ur ≤ 0 on PT and


rur = f(u, ur) in QT . (52.28)

Then ur ≤ 0 in QT . If ur(·, 0) ≡ 0, then ur < 0 in QT and, assuming R < ∞,ur(R, t) < 0 for t ∈ (0, T ).

Proof. For λ > 0 large enough, the function w := ure−λt solves the equation

wt − wrr = bwr + cw a.e. in QT ,

with b = fξ(u, ur)+ n−1r and c = fu(u, ur)−λ− n−1

r2 ≤ 0. Assume for contradictionthat supQT

w > 0. Set m = supQTw/2, z = w −m, and choose r0 ∈ (0, R) such

that w ≤ m in [0, r0]× [0, T ]. Then, for some K > 0, we have

zt − zrr = bzr + c(z + m) ≤ K|zr|+ cz a.e. in (r0, R)× (0, T ).

Using Proposition 52.8 we infer that z ≤ 0 in (r0, R) × [0, T ], contradicting thedefinition of m. Consequently, ur ≤ 0.

Finally, to show that ur < 0 it is sufficient to use Proposition 52.7 with Ω =(ε, R), where 0 < ε < R < ∞, R ≤ R.

Proof of Proposition 52.17. Let us first verify assertion (i). We know that uis radial and u satisfies (52.28) with f(u, ur) = F (u, (ur, 0, . . . , 0)).

520 Appendices

We have ur ∈ BC([0, R] × [0, τ ]), 0 < τ < T (with [0, R] replaced by [0,∞)if Ω = R

n). Also, by Remark 48.3(i), we have ur ∈ W 2,1;2loc (QT ). Since u0,r ≤

0, ur(0, t) = 0, and ur(R, t) ≤ 0 due to u ≥ 0, the assertion is then a directconsequence of Lemma 52.18.

For the proof of assertions (ii) and (iii), we proceed in several steps.

Step 1. Define D(Ω) = v ∈ D(Ω) : v ≥ 0, v is radial nonincreasing andLq(Ω) = v ∈ Lq(Ω) : v ≥ 0, v is radial nonincreasing for 1 ≤ q ≤ ∞. It isnot too difficult to show that if u0 ∈ Lq(Ω) and v ∈ D(Ω), then the convolutionproduct u0 ∗ v is radial nonincreasing, hence belongs to Lq(Ω). The same is trueif v is replaced by Gt, t > 0.

Step 2. By standard arguments of truncation and convolution with a mollifier,using Step 1, one shows that D(Ω) is dense in Lq(Ω).

Step 3. We claim that for u0 satisfying the assumption of (ii) or (iii), the functionz(x, t) := e−tAu0 is radial nonincreasing. If Ω = R

n, then this follows from Step 1,since z(t) = Gt ∗ u0. If Ω = BR, then this is true for u0 ∈ D(Ω) by assertion (i),and the general case follows from the density property of Step 2.

Step 4. First consider case (ii) with f ′ ≥ 0, or case (iii). The solution u isconstructed as the fixed point of a suitable contraction mapping (cf. the proofof Theorem 15.2 and Remark 51.11). Consequently, u is the limit of a sequenceuk+1 = Φu0(uk) := e−tAu0 +

∫ t

0 e−(t−s)Af(u(s)) ds. The conclusion then followsfrom the fact that the operator Φu0 preserves the radial nonincreasing property,due to f ′ ≥ 0.

Step 5. In case (ii) for general f of class C1, let M := supQTu and set f(s) :=

f(s) + λs, where λ > 0 is chosen so large that f ′ ≥ 0 on [0, M ]. Noting that u

solves ut − (∆ − λ)u = f(u), the conclusion follows from the argument in Step 4,where e−tA is replaced with e−λte−tA.

Finally, let u0 ≡ 0. Since, given δ > 0, u(δ) ∈ BC1(Ω) and u = 0 on ∂Ω,inequality (52.27) follows from (i).

52.6. Monotonicity of solutions in time

We give two results which are useful to guarantee the monotonicity of solutions intime.

Proposition 52.19. Let Ω ⊂ Rn be a uniformly regular domain of class C2, let

F = F (s, ξ) : R×Rn → R be a C1-function, and consider problem (34.1). Assume

that u0 ∈ BC(Ω) ∩H2loc(Ω) satisfies u0 = 0 on ∂Ω and

∆u0 + F (u0,∇u0) ≥ 0 a.e. in Ω.

If F depends on ξ, assume in addition that u0 ∈ BC1(Ω). Then ut ≥ 0 in QT ,where T := Tmax(u0).

Proof. In the case when F depends on ξ, first recall that problem (34.1) is well-posed in X = u ∈ BC1(Ω) : u = 0 on ∂Ω.

By comparing u with the subsolution u(x, t) := u0(x) via Proposition 52.10, weobtain u ≥ u0 in QT .

Now fix h ∈ (0, T ) and put v(t) := u(t + h). Since v(0) = u(h) ≥ u0, we inferfrom Proposition 52.6 that v ≥ u on (0, T −h). The result then follows by dividingby h and letting h → 0.

In case Ω is bounded and the nonlinearity depends only on u, the followingalternative approach guarantees monotonicity of solutions in time under muchweaker regularity on the initial data. We say that u0 ∈ L∞(Ω) satisfies

∆u0 + f(u0) ≥ 0, x ∈ Ω,

u0 ≤ 0, x ∈ ∂Ω

(52.29)

in the very weak sense if, for all 0 ≤ ψ ∈ C2(Ω) such that ψ = 0 on ∂Ω, thereholds ∫

Ω

u0∆ψ + f(u0)ψ

dy ≥ 0. (52.30)

(Of course, this is satisfied in particular if u0 belongs to H2 ∩H10 (Ω) and verifies

∆u0 + f(u0) ≥ 0 a.e. in Ω.)

Proposition 52.20. Assume that Ω is a bounded domain of class C2+α for someα ∈ (0, 1) and consider problem (14.1) with f ∈ C1(R). If u0 ∈ L∞(Ω) satisfies(52.29) in the very weak sense, then ut ≥ 0 in QT , where T := Tmax(u0).

Proof. Step 1. We claim that u ≥ u0 in QT .For 0 ≤ t < T , set v(t) := u(t)− u0, c(x, t) = (f(u)− f(u0))/(u − u0) (defined

to be 0 whenever the denominator vanishes), and notice that c ∈ L∞(QT ).For given 0 < τ < T , let 0 ≤ ϕ ∈ C2(Ω× [0, τ ]) be such that ϕ = 0 on ∂Ω× [0, τ ]

and ϕ(τ) = 0. For each 0 < t < τ , by integrating by parts and using (52.30) withψ =

∫ τ

tϕ as a test-function, we obtain

−∫

Ω

(vϕ)(t) dx =

∫ τ

t

∫Ω

(vϕt + utϕ

)dx ds =

∫ τ

t

∫Ω

(vϕt + (∆u + f(u))ϕ

)dx ds

=∫ τ

t

∫Ω

v(ϕt + ∆ϕ) + f(u)ϕ + u0∆ϕ

dx ds

≥∫ τ

t

∫Ω

v(ϕt + ∆ϕ) + (f(u)− f(u0))ϕ

dx ds

=∫ τ

t

∫Ω

v(ϕt + ∆ϕ + cϕ)

dx ds.

522 Appendices

Letting t → 0, hence ‖u(t)− u0‖1 → 0 (due to ‖u(t)− e−tAu0‖∞ → 0), it followsthat ∫ τ

t

∫Ω

v(ϕt + ∆ϕ + cϕ)

dx ds ≤ 0.

By Proposition 52.13, we deduce that v ≥ 0, hence the claim.Step 2. As before, we fix h ∈ (0, T ) and let v(t) := u(t + h). Since

u, v ∈ C2,1(Ω× (0, T − h)) ∩ L∞loc(Ω× [0, T − h)) ∩ C([0, T − h), L1(Ω))

are classical solutions of the first two equations in (14.1) on (0, T − h), we deducefrom Proposition 52.13(i) and Remark 52.12(i) that v ≥ u on (0, T −h). The resultfollows by dividing by h and letting h → 0.

52.7. Systems and nonlocal problems

We first give extensions of some of the preceding results to systems of cooperativetype.

Proposition 52.21. Let 0 < T < ∞, Ω be an arbitrary domain in Rn, d1, d2 > 0,

and aij ∈ L∞(QT ), i, j ∈ 1, 2, with a12, a21 ≥ 0. Assume that for i = 1, 2,the function wi satisfies wi ∈ C(Ω × (0, T )) ∩ C([0, T ), L2

loc(Ω)), supQTwi < ∞,

∂twi,∇wi, D2wi ∈ L2

loc(QT ). If w1, w2 ≤ 0 on PT and

∂tw1 − d1∆w1 ≤ a11w1 + a12w2 a.e. in QT ,∂tw2 − d2∆w2 ≤ a21w1 + a22w2 a.e. in QT ,

then

w1, w2 ≤ 0 in QT .

Proof. Let ε > 0, λ = 2 max0≤i,j≤2

supQT

aij , and set

zi = wie−λt − εψ,

where ψ defined in (52.12) with M = n. Since ∆ψ − ψt ≤ 0 by (52.6), it followsthat a.e. in QT , there holds

∂tz1 − d1∆z1 = e−λt(∂tw1 − d1∆w1 − λw1) + ε(∆ψ − ψt)

≤ e−λt((a11 − λ)w1 + a12w2

)≤ (a11 − λ)z1 + a12z2 + ε(a11 + a12 − λ)ψ ≤ (a11 − λ)z1 + a12z2

and similarly,∂tz2 − d2∆z2 ≤ a21z1 + (a22 − λ)z2.

Arguing as in the proof of Proposition 52.8, it follows that

12

d

dt

∫Ω

(z1,+)2(t) dx

≤ −d1

∫Ω

|∇(z1,+)|2 dx +∫

Ω

(a11 − λ)(z1,+)2 dx +∫

Ω

a12 z2 z1,+ dx

≤∫

Ω

a12 z2,+ z1,+ dx ≤ C

∫Ω

((z1,+)2 + (z1,+)2

)dx,

(52.31)

where we used a11 − λ ≤ 0 and a12 ≥ 0. Similarly, we get

12

d

dt

∫Ω

(z2,+)2(t) dx ≤ C

∫Ω

((z1,+)2 + (z1,+)2

)dx. (52.32)

Adding up (52.31) and (52.32), integrating, and using z1,+(0) = z2,+(0) = 0, weinfer that z1,+ = z2,+ = 0 in QT and the conclusion follows by letting ε → 0.

By arguing similarly as in the proof of Proposition 52.10, one obtains a com-parison principle for cooperative systems of the form

∂tui − di∆ui − fi(u1, u2) = 0, i = 1, 2. (52.33)

Proposition 52.22. Let 0 < T < ∞, Ω be an arbitrary domain in Rn, and let

fi = fi(u1, u2) : R2 → R, i = 1, 2, be C1-functions such that

∂u2f1 ≥ 0, ∂u1f2 ≥ 0. (52.34)

Let u = (u1, u2), where ui ∈ C(Ω × (0, T )) satisfy ui ∈ L∞(QT ), ui ∈ C([0, T ),L2

loc(Ω)), and ∂tui,∇ui, D2ui ∈ L2

loc(QT ). Finally, let v satisfy the same hypothe-ses as u. If, for i = 1, 2, we have ui ≤ vi on PT and

∂tui − di∆ui − fi(u1, u2) ≤ ∂tvi − di∆vi − fi(v1, v2) a.e. in QT ,

thenui ≤ vi in QT , i = 1, 2.

Remarks 52.23. (i) The cooperativity assumption (52.34) (or a12, a21 ≥ 0 inProposition 52.21) is essential to ensure the order-preserving character of system(52.33), as shown by the following simple example. Consider system (52.33) underhomogeneous Dirichlet boundary conditions, with f1(u, v) = −v, f2(u, v) = 0. Ifwe take u0 = 0 and v0 ≥ 0, v0 ≡ 0, then, by the strong maximum principle, wehave v > 0, hence u < 0, in Ω× (0,∞). Therefore the order with the solution (0, 0)at t = 0 is not preserved.

(ii) For system (52.33) with homogeneous Dirichlet boundary condition, underassumption (52.34), the analogues of Propositions 52.19 and 52.20 guaranteeingtime-monotonicity of solutions can be established by simple modifications of theproofs.

We next turn to nonlocal problems (with space or time integral nonlinearities).

524 Appendices

Proposition 52.24. Let 0 < T < ∞, Ω be an arbitrary bounded domain in Rn,

and a, b, k ∈ L∞(QT ), with b, k ≥ 0. Assume that the function w ∈ C(Ω× (0, T ))∩C([0, T ), L2(Ω)) satisfies supQT

w <∞,

∂tw,∇w, D2w ∈ L2loc(Ω× (0, T )). (52.35)

If w ≤ 0 on PT and either

∂tw −∆w ≤ aw + b

∫Ω

k(y, ·)w(y, ·) dy a.e. in QT , (52.36)

or

∂tw −∆w ≤ aw + b

∫ t

0

k(·, s)w(·, s) ds a.e. in QT , (52.37)

thenw ≤ 0 in QT .

Proof. Our assumptions imply w+ ∈ C([0, T ), L2(Ω))∩C1((0, T ), L2(Ω)), w+(0)= 0 and, for a.e. t ∈ (0, T ), w+(t) ∈ H1

0 (Ω). Moreover, for a.e. t ∈ (0, T ), we have∆w(·, t) ∈ L2(Ω), and ∇(w+)(·, t) = χw>0∇w(·, t).

In the case of (52.36), by using b, k ≥ 0 and the Cauchy-Schwarz inequality, weobtain

12

d

dt

∫Ω

(w+)2(t) dx ≤ −∫

Ω

|∇(w+)|2 dx +∫

Ω

a(w+)2 dx +∫

Ω

bw+ dx

∫Ω

kw+ dy

≤ C

∫Ω

(w+)2 dx.

In the case of (52.37), we obtain

12

d

dt

∫Ω

(w+)2(t) dx ≤ −∫

Ω

|∇(w+)|2 dx +∫

Ω

a(w+)2 dx

+∫

Ω

bw+

(∫ t

0

kw+ ds)dx

≤∫

Ω

a(w+)2 dx +∫

Ω

b2(w+)2 dx + T

∫ t

0

∫Ω

k2(w+)2 dx ds.

The function φ(t) :=∫ t

0

∫Ω(w+)2 dx ds thus satisfies φ′′ ≤ C(φ + φ′) and φ, φ′ ≥ 0,

hence[φ2 + (φ′)2]′ = 2(φ + φ′′)φ′ ≤ C[φ2 + (φ′)2], 0 < t < T,

with φ(0) = φ′(0) = 0.In both cases, by integration, we conclude that w+ = 0 in QT .

As a consequence of Proposition 52.24 we obtain in particular the followingcomparison principle. The proof is similar to that of Proposition 52.10.

Proposition 52.25. Let 0 < T < ∞, Ω be an arbitrary bounded domain in Rn,

and let f : R → R and g = g(s, z) : R2 → R be C1-functions, with either f ′, ∂zg ≥

0 or f ′, ∂zg ≤ 0. Let u ∈ C(Ω×(0, T )) satisfy u ∈ L∞(QT ), u ∈ C([0, T ), L2loc(Ω)),

and ut,∇u, D2u ∈ L2loc(Ω × (0, T )), and let v satisfy the same hypotheses as u.

Finally, denote

I(u, t) :=∫

Ω

f(u(y, t)) dy (resp., I(u, t) :=∫ t

0

f(u(y, s)) ds).

If u ≤ v on PT and

ut −∆u− g(u, I(u, ·)

)≤ vt −∆v − g

(v, I(v, ·)

)a.e. in QT ,

thenu ≤ v in QT .

Remarks 52.26. (i) The positivity assumption on b, k is essential for the validityof the nonlocal maximum principle in Proposition 52.24, as shown by the followingexample from [525]: The function w(x, t) = x2 − t satisfies

wt − wxx = −3 ≥ −18∫ 1

−1w(y, t) dy, 1 < x < 1, 0 < t < 1/4,

w(±1, t) = 1− t ≥ 0, 0 < t < 1/4,

w(x, 0) = x2 ≥ 0, 1 < x < 1,

⎫⎪⎪⎬⎪⎪⎭ (52.38)

but w(0, t) = −t < 0.(ii) Assumption (52.35) in Proposition 52.24 can be weakened to ∂tw,∇w,

D2w ∈ L2loc(QT ) (and similarly in Proposition 52.25). To see this it suffices to

replace w in the proof by z := w − εeλt with λ > 0 large (using the fact thatz+ = 0 near the boundary similarly as in the proof of Proposition 52.8), and thenlet ε → 0.

In the case of nonlocal problems in unbounded domains, we need a differentstatement.

Proposition 52.27. Let 0 < T < ∞, Ω be an arbitrary domain in Rn, a, b ∈

L∞(QT ), and k ∈ L∞(Ω), with b, k ≥ 0. Assume that the function w ∈ C2,1(QT )∩C(QT ) satisfies w ≤ 0 on PT ,

∂tw −∆w ≤ aw + b

∫Ω

k(y)w(y, ·) dy a.e. in QT , (52.39)

andw ∈ C([0, T ), L1(Ω)),

∫Ω

k(y)w(y, 0) dy < 0. (52.40)

526 Appendices

Thenw < 0 in QT .

Proof. Denote I(t) :=∫Ω k(y)w(y, t) dy and let 0 < τ < T . We claim that if

I(t) ≤ 0, 0 ≤ t ≤ τ, (52.41)

thenw < 0, x ∈ Ω, 0 < t ≤ τ. (52.42)

Indeed, let z := eλtw with λ := − infQT a and a := a + λ ≥ 0. By (52.41), (52.39),we have zt −∆z ≤ az in QT . Since z ≤ 0 on PT , it follows from Proposition 52.8that z ≤ 0, hence zt −∆z ≤ 0 in QT . Since z(0) ≡ 0, the claim then follows fromthe standard strong maximum principle (see [429], or use Proposition 52.7 in anysmooth bounded subdomain of Ω).

Now, by (52.40), the function I(t) is continuous, with I(0) < 0. Therefore,(52.41) is true for small τ > 0. Let

T0 := supτ ∈ (0, T ) : (52.41) (hence (52.42)) is true

and assume for contradiction that T0 < T . Then (52.41) and (52.42) hold forτ = T0, hence in particular w(·, T0) < 0. Consequently, I(T0) < 0, so that (52.41)holds for some τ > T0, contradicting the definition of T0. This proves the result.

52.8. Zero number

Zero number arguments can be viewed as a sophisticated form of the maximumprinciple. Although they are restricted to one-dimensional or radially symmetricproblems, they represent a very powerful tool.

The zero number of a function ψ ∈ C((0, R)) is defined as the number of signchanges of ψ in (0, R);

z(ψ) = z[0,R](ψ) = supk ∈ N : there are 0 < x0 < x1 < · · · < xk < R

such that ψ(xi)ψ(xi+1) < 0 for 0 ≤ i < k.

Let BR = x ∈ Rn : |x| < R, t1 < t2, q ∈ L∞(BR, (t1, t2)), u ∈ C(BR ×

[t1, t2]) ∩W 2,1;∞(BR × (t1, t2)) and

ut −∆u = qu a.e. in BR × (t1, t2). (52.43)

Assume that q(·, t) and u(·, t) are radially symmetric for all t, hence q(x, t) =Q(|x|, t) and u(x, t) = U(|x|, t). Then

Ut − Urr −n− 1

rUr = QU, r ∈ (0, R), t ∈ (t1, t2), (52.44)

and Ur(0, t) = 0 for all t ∈ (t1, t2).

Theorem 52.28. Let q, u be as above, u ≡ 0, and either U(R, t) = 0 for allt ∈ [t1, t2] or U(R, t) = 0 for all t ∈ [t1, t2]. Let z = z[0,R] denote the zero numberin (0, R). Then(i) z(U(·, t)) <∞ for all t ∈ (t1, t2),(ii) the function t → z(U(·, t)) is nonincreasing,(iii) if U(r0, t0) = Ur(r0, t0) = 0 for some r0 ∈ [0, R] and t0 ∈ (t1, t2), thenz(U(·, t)) > z(U(·, s)) for all t1 < t < t0 < s < t2.

Proof. If U(R, t) = 0 for all t, then the assertion follows from [127, Theorem 2.1].If U(R, t) = 0 for all t, then we may assume U(R, t) > 0 for all t. Fix ε ∈ (0, R)

such that U(r, t) > ε for all r ∈ [R − ε, R] and t ∈ [t1, t2]. Let V = V (r) be thesolution of

Vrr +n− 1

rVr = 0 in [R− ε, R + ε], V (R + ε) = 0, Vr(R + ε) = −1,

and notice that V (r) ≥ ε for r ≤ R. Choose ϕ ∈ C∞([0, R+ε]) such that 0 ≤ ϕ ≤ 1,ϕ ≡ 1 on [0, R− ε], ϕ ≡ 0 on [R, R + ε], and set

U(r, t) = ϕ(r)U(r, t) + (1− ϕ(r))V (r), r ∈ [0, R + ε], t ∈ [t1, t2],

Q(r, t) =

⎧⎪⎨⎪⎩Q(r, t) if r ∈ [0, R− ε],1U

(Ut − Urr − n−1

r Ur

)if r ∈ (R− ε, R),

0 if r ∈ [R, R + ε].

Then U solves (52.44) with Q replaced by Q and R by R + ε, U(R + ε, t) = 0 andthe assertion follows from z[0,R](U(·, t)) = z[0,R+ε](U(·, t))

Remarks 52.29. (i) The assertion of Theorem 52.28 remains true for more gen-eral problems of the form

ut −∆u = qu + bx · ∇u,

where b ∈ W 1,∞(BR × (t1, t2)), b(x, t) = B(|x|, t). This follows from the fact thatthe function v(x, t) := e

12

∫ |x|0 B(ξ,t)ξ dξu(x, t) solves a problem of the form (52.43).

(ii) If n = 1, then a more general statement (allowing Dirichlet, Neumannor periodic boundary conditions and more general coefficients of the differentialoperators) can be found in [30]. In particular, the arguments in [30] guarantee thatif x1 < x2, t1 < t2 and u = U ∈ C([x1, x2]× [t1, t2]) is a solution of

ut − a(x, t)uxx = b(x, t)ux + c(x, t)u in (x1, x2)× (t1, t2),

wherea > 0, a, a−1, at, ax, axx, b, bt, bx, c ∈ L∞,

528 Appendices

and, for any i ∈ 1, 2,

either u(xi, t) = 0 for t ∈ [t1, t2],

or u(xi, t) = 0 for t ∈ [t1, t2],

then statements (i)–(iii) in Theorem 52.28 hold with [0, R] replaced by [x1, x2]. Letus mention that in the case of Neumann boundary conditions one has to assumea ≡ 1 and b ≡ 0.

53. Appendix G: Dynamical systems

In this section we collect some basic definitions and properties of dynamical sys-tems. Since the statements are usually proved only in the dissipative case in theliterature (see [267], for example), we also provide detailed proofs.

Definition 53.1. Let (X, d) be a complete metric space and τ : X → (0,∞] belower semicontinuous. A mapping ϕ : X × [0,∞) → X defined for all (u, t) withu ∈ X and t ∈ [0, τ(u)) is called a (local) dynamical system on X if(i) ϕ(u, ·) : [0, τ(u)) → X is continuous,(ii) ϕ(·, t) : X → X is continuous at u for all u ∈ X and t < τ(u),(iii) ϕ(u, 0) = u for all u ∈ X ,(iv) τ(ϕ(u, s)) = τ(u)− s and ϕ(u, t+ s) = ϕ(ϕ(u, s), t) for all u ∈ X , s ∈ [0, τ(u))and t ∈ [0, τ(u)− s).

In our applications, X is typically a Banach space in which the studied problemis well-posed (or just the positive cone of such space), τ(u) is the maximal time ofexistence of the solution with initial data u and ϕ(u, t) is this solution at time t.Note that for most of our assertions, ϕ need not be continuous with respect to tat t = 0 so that we can also choose X = L∞(Ω), for example.

Given u ∈ X , the mapping ϕu : [0, τ(u)) : t → ϕ(u, t) is called the trajectoryemanating from u. It is global if τ(u) = ∞. A point u ∈ X is called anequilibrium if τ(u) = ∞ and ϕ(u, t) = u for all t ≥ 0. We denote by S the set ofall equilibria.

If τ(u) =∞, then we define the ω-limit set of ϕu by

ω(ϕu) = ω(u) := v ∈ X : there exist tk → +∞such that ϕ(u, tk) → v as k →∞.

(53.1)

It is easy to see that

ω(u) =⋂s>0

⋃t≥s

ϕ(u, t) (53.2)

and ω(ϕ(u, t)) = ω(u) for all t > 0.

53. Appendix G: Dynamical systems 529

Proposition 53.2. Let τ(u) = ∞ and v ∈ ω(u). Then ϕ(v, t) ∈ ω(u) for allt ∈ [0, τ(v)).

Proof. There exist tk → ∞ such that ϕ(u, tk) → v. Given t ∈ [0, τ(v)), setτk := tk + t. Then ϕ(u, τk) = ϕ(ϕ(u, tk), t)→ ϕ(v, t), hence ϕ(v, t) ∈ ω(u).

Proposition 53.3. Assume

τ(u) = ∞ and⋃t≥0

ϕ(u, t) is relatively compact in X. (53.3)

Then τ(v) = ∞ for all v ∈ ω(u), ω(u) is compact, connected, nonempty, invariant(that is ϕ(ω(u), t) = ω(u) for all t > 0) and d(ϕ(u, t), ω(u)) → 0 as t→∞.

Proof. The set K := ϕ(u, t) : t ≥ 0 is compact and ω(u) ⊂ K. The set ω(u) isclosed due to (53.2), hence compact.

Choose tk → ∞. Then ϕ(u, tk) is relatively compact, hence passing to asubsequence we may assume ϕ(u, tk) → v. Consequently, v ∈ ω(u) and ω(u) isnonempty.

Fix v ∈ ω(u). Proposition 53.2 guarantees ϕ(v, t) ∈ ω(u) for all t ∈ [0, τ(v)).Assume on the contrary that τ(v) < ∞. Choose tk ∈ [0, τ(v)), tk → τ(v). Thenϕ(v, tk) is relatively compact and passing to a subsequence we may assumeϕ(v, tk)→ v∞. Then τ(ϕ(v, tk)) = τ(v)− tk → 0 and τ(v∞) > 0 which contradictsthe lower semicontinuity of τ . Consequently, τ(v) =∞.

Due to Proposition 53.2, in order to show the invariance of ω(u) it is sufficientto prove

ω(u) ⊂ ϕ(ω(u), t) for all t > 0. (53.4)

Fix v ∈ ω(u), t > 0 and tk →∞ such that ϕ(u, tk) → v. Set τk := tk − t. Passingto a subsequence we may assume ϕ(u, τk)→ w ∈ ω(u). Then

ϕ(w, t) = ϕ(

limk→∞

ϕ(u, τk), t)= lim

k→∞ϕ(u, tk) = v,

which proves (53.4).Next we show that d(ϕ(u, t), ω(u)) → 0 as t → ∞. Assume on the contrary

that there exist tk → ∞ and ε > 0 such that d(ϕ(u, tk), ω(u)) ≥ ε. Passing to asubsequence we may assume ϕ(u, tk)→ v ∈ ω(u) which yields a contradiction.

For any s > 0, the set⋃ϕ(u, t) : t ≥ s is connected and relatively compact,

hence its closure is connected and compact. Due to (53.2), ω(u) is a decreasingintersection of connected compact sets, hence ω(u) is connected.

A continuous function E : X → R is called a Lyapunov function for ϕ ifE(ϕ(u, t)) ≤ E(u) for all u ∈ X and t ∈ [0, τ(u)).

530 Appendices

Proposition 53.4. Let E be a Lyapunov function and (53.3) be true. Then thelimit e := limt→∞ E(ϕ(u, t)) exists and E(v) = e for all v ∈ ω(u).

Proof. The function t → E(ϕ(u, t)) is nonincreasing and bounded since the set⋃ϕ(u, t) : t ≥ 0 is relatively compact. Hence e := limt→∞ E(ϕ(u, t)) exists.If v ∈ ω(u), then there exist tk → ∞ such that ϕ(u, tk) → v. Consequently,

E(ϕ(u, tk)) → E(v) = e.

A Lyapunov function E is called a strict Lyapunov function if the follow-ing condition is satisfied: If E(ϕ(u, t)) = E(u) for all t ∈ [0, τ(u)), then u is anequilibrium.

The following two useful results are versions of Lasalle’s invariance principle.

Proposition 53.5. Let E be a strict Lyapunov function and (53.3) be true. ThenS is a closed nonempty set and d(ϕ(u, t), S) → 0 as t → ∞. In particular, ω(u)consists of equilibria.

Proof. The continuity of ϕ guarantees that S is closed and Proposition 53.3 showsthat ω(u) = ∅. Fix v ∈ ω(u) and t ≥ 0. Then τ(v) = ∞ and ϕ(v, t) ∈ ω(u) due toProposition 53.3, hence Proposition 53.4 implies E(ϕ(v, t)) = E(v). Consequently,v ∈ S.

Proposition 53.6. Assume τ(u) =∞, tk →∞ and ϕ(u, tk) → v. Let there exista strict Lyapunov function E. Then v ∈ S.

Proof. The proof of Proposition 53.4 shows that e := limk→∞ E(ϕ(u, tk)) existsand E(v) = e. Fixing t ∈ [0, τ(v)), the continuity of ϕ implies ϕ(u, tk+t)→ ϕ(v, t).As above, e := limk→∞ E(ϕ(u, tk + t)) exists and E(ϕ(v, t)) = e. Fixing k ∈ N

there exists j ∈ N such that tk+j ≥ tk + t ≥ tk, hence

E(ϕ(u, tk+j)) ≤ E(ϕ(u, tk + t)) ≤ E(ϕ(u, tk)),

thus e = e and E(ϕ(v, t)) = E(v). Consequently, v ∈ S.

Example 53.7. Consider problem (17.1) with p > 1 and λ ∈ R, and let q ∈[max(qc, p+1),∞). By Example 51.28, we know that this problem generates a dy-namical system on the space X := H1

0∩Lq(Ω). Moreover, by (17.7) in Lemma 17.5,the energy functional E(u) defined in (17.6) is a strict Lyapunov functional.Now assume Ω bounded and let u0 ∈ L∞(Ω) be such that Tmax(u0) = ∞ andsupt>0 ‖u(t)‖∞ <∞. Then, as a consequence of parabolic estimates and Proposi-tion 53.5, for each τ > 0, the set u(t) : t ≥ τ is relatively compact in X and theω-limit set ω(u0) (in the X-topology) is nonempty and consists of (classical) equi-libria. Moreover, by smoothing effects, the convergence in the definition of ω(u0)actually takes place (for instance) in C1+β(Ω) for each β ∈ (0, 1).

By similar arguments, the above facts remain true for the more general problem(14.1) with f ∈ C1 if we replace the last integral in (17.6) by

∫Ω

F (u) dx, where

53. Appendix G: Dynamical systems 531

F (u) =∫ u

0f(s) ds, and X by H1 ∩C0(Ω). (Note that, as far as Propositions 53.3–

53.6 are concerned, one could alternatively work with X := H1∩L∞(Ω): Althoughthe continuity property at t = 0 in Definition 53.1(i) is not true, this is of noimportance in those results.)

Under an additional time monotonicity assumption, one obtains the convergenceof the trajectory to a single equilibrium. See Remark 19.13 for other conditionsguaranteeing convergence in the case of dynamical systems generated by parabolicdifferential equations.

Proposition 53.8. Let (X,≤) be an ordered Banach space with a closed positivecone X+ := u ∈ X : u ≥ 0. Assume that (53.3) is true and the trajectory ϕu

is nondecreasing, that is ϕ(u, t1) ≤ ϕ(u, t2) whenever t1 ≤ t2. Then ω(u) is asingleton contained in S.

Proof. Proposition 53.3 guarantees that ω(u) is nonempty and invariant. Letv1, v2 ∈ ω(u). Then there exist t1k → ∞ and t2k → ∞ such that ϕ(u, tik) → vi,i = 1, 2. Without loss of generality we may assume t1k < t2k < t1k+1 for all k. Thenϕ(u, t1k) ≤ ϕ(u, t2k) ≤ ϕ(u, t1k+1) and passing to the limit we obtain v1 ≤ v2 ≤ v1,hence v1 = v2. Consequently, ω(u) is a singleton. Since it is an invariant set, wehave ω(u) ⊂ S.

Remark 53.9. For many dynamical systems generated by parabolic differentialequations, the compactness assumption (53.3) in Proposition 53.8 can be replacedby a weaker boundedness assumption. In fact, the monotonicity of the solutionin time usually enables one to pass to the limit and conclude that the limit is astationary solution. For example one can often use the following lemma. In the caseof monotonicity in space, similar arguments are used in the proofs of Theorems 8.3and 21.10(ii).

Lemma 53.10. Let f : R → R be locally Holder continuous, let Ω be an arbitrarydomain in R

n, and set Q := Ω× (0,∞). Let u ∈ C2,1(Q) satisfy

ut −∆u = f(u), (x, t) ∈ Q, (53.5)

andsupQ|u| <∞.

Assume that ut ≥ 0 in Q, and let v(x) := limt→∞ u(x, t). Then v is a boundedclassical solution of

−∆v = f(v), x ∈ Ω. (53.6)

Proof. Let wj(x, t) = u(x, t + j). The Lp and Schauder interior parabolic esti-mates guarantee that the sequence wj is relatively compact in C2,1(K) for eachcompact subset K of Q. It follows that some subsequence of wj converges to a

532 Appendices

classical solution w = w(x, t) of (53.5). But it is clear that v(x) = w(x, t) for each(x, t) ∈ Q. The conclusion follows.

Alternative proof. Let ϕ ∈ D(Ω). For each t > 0, multiplying by ϕ and inte-grating by parts yields

[∫Ω

uϕdx]t+1

t+∫ t+1

t

∫Ω

(u∆ϕ + f(u)ϕ

)dx ds = 0.

Passing to the limit by dominated convergence as t →∞, we obtain∫Ω

(v∆ϕ + f(v)ϕ

)dx = 0.

It follows that v is a distributional solution of (53.6), hence a classical solution bystandard elliptic regularity results (cf. Remarks 47.4).

Let us ∈ S. The domain of attraction of us is the set

D = D(us) := u : τ(u) = ∞ and ϕ(u, t) → us as t→∞.

We say that us is asymptotically stable if D(us) contains a neighborhood of us.If us is asymptotically stable, then D(us) is obviously open. If u ∈ ∂D(us), thenthe continuity of ϕ implies ϕ(u, t) ∈ ∂D(us) for all t ∈ (0, τ(u)).

Let u, v ∈ S, u = v. A function ψ : R → X is called a connecting (orheteroclinic) orbit between u and v if limt→−∞ ψ(t) = u, limt→+∞ ψ(t) = v andψ(t) = ϕ(ψ(s), t− s) for all −∞ < s < t < ∞.

54. Appendix H: Methodological notes

In this orientation section, we shall summarize the different methods employedin this monograph for each of the main questions that we address. Of course,there exist other important methods which are frequently exploited for relatedquestions and which are not represented in this book for various reasons. Forexample, in the elliptic part we only use the simplest variational methods to provethose existence results which are needed in the parabolic part and we do not payattention to linking or concentration compactness. Notice also that our summarydoes not even contain all important methods used in this monograph. For example,matched asymptotics is used in Section 29 in the study of decay and grow-up rateswhich is not a central theme for us. On the other hand, matched asymptotics playsa crucial role in several fundamental papers devoted to our main questions (likeblow-up rates or blow-up profiles, see [277], [522], [275], [276], for example) butthe corresponding rigorous proofs are too long and technical and lie beyond thescope of this book.

54. Appendix H: Methodological notes 533

Before providing the summary, let us briefly classify the main tools and tech-niques that are used throughout the book. We point out that, of course, many ofthese techniques are not specific to the field of superlinear elliptic and parabolicproblems but are classical in various areas of PDE’s.

A. Background tools

A1. Tools from Functional Analysis (functional spaces and inequalities, inter-polation)

A2. Tools from ODE’s (differential and integral inequalities, phase plane anal-ysis)

A3. Linear elliptic and parabolic estimates (often used to obtain compactnessproperties)

A4. Tools from Dynamical Systems (ω-limits, Lyapunov functionals)

B. Main classes of techniques

B1. Comparison techniques (based on maximum principles, including movingplanes and zero-number)

B2. Test-function and multiplier techniques7 (in particular including varia-tional and energy methods). In the parabolic case this often leads to adifferential inequality for a functional of the solution

B3. Semigroup techniques (relying on the variation-of-constants formula)B4. Rescaling procedures (often leading to the use of a nonlinear Liouville-type

theorem via a contradiction argument)8

B5. Bootstrap and iteration procedures

C. Some other techniques

C1. Changes of dependent and/or independent variables (cf., e.g., similarityvariables, Hopf-Cole transformation, conversion to a problem with absorp-tion, . . . )

C2. Differentiation of the PDE (cf., e.g., Bernstein-type techniques, auxiliaryfunctions J , . . . )

C3. Monotonicity techniques: use of monotonicity properties of solutions (intime or in space — usually obtained via a maximum principle), monotoneapproximation (cf. complete blow-up, threshold trajectories)

C4. Doubling arguments

7By a test-function technique, we usually understand the space or space-time integrationof the PDE after multiplication by a function independent of the solution itself, such as the firsteigenfunction for instance. In a multiplier technique, the function may depend on the solution,e.g. a power of the solution.

8Another aspect of the concept of scaling is the existence of self-similar solutions —cf. also similarity variables in C1.

534 Appendices

C5. Duality arguments

We now turn to the more detailed summary of methods. For a given ques-tion, the choice of the applicable (and more appropriate) methods will dependon the particular properties of the problem: scale invariance, variational struc-ture, monotonicity, convexity or boundedness of the domain, regularity assumedon solutions, . . .

For some partial comparative discussion, see e.g. the beginning of sections 10–13; Remark 26.5 and the end of Remark 26.12; the paragraph after Theorem 27.2;Remarks 31.5, 31.18(i) and the beginning of Subsection 31.4.

We stress that the following list, which is mainly intended as a help and aguideline for readers, is necessarily schematic. Certain proofs may sometimes in-volve combinations of several methods, or some ad hoc arguments which do notappear in the list. On the other hand, some items below may partially overlap.The places where each method is used are mentioned in italic between brackets.

I. METHODS FOR ELLIPTIC PROBLEMS

M1. Methods to prove existence of solutions

M1.1. Variational methods(a) Minimisation under constraint [Section 6](b) Minimax methods [Section 7]

M1.2. ODE methods [Section 9]M1.3. A priori estimates and topological degree argument [Corollary 10.3 and

cf. M4]M1.4. Dynamical methods

(a) Method based on a priori estimates of global solutions and thresholdtrajectories [Remark 28.8(ii)]

(b) Stabilization of monotone bounded solutions [Theorem 43.1(iii)]

M2. Methods to prove nonexistence of solutions

Note: the methods which are mainly motivated by Liouville-type results appearonly in M8.1–M8.3 below.

M2.1. Variational identities of Pohozaev-type [Corollary 5.2, Proposition 25.4,Theorem 31.3]

M2.2. Multiplication by the first eigenfunction [Remark 6.3]M2.3. ODE techniques [Section 9]

Note: the methods in M2.1 and M2.3 may be combined with symmetryresults [Remark 6.9(i)]

M2.4. Maximum principle [Proposition 40.8]

M3. Methods to study regularity and singularities of solutions

M3.1. To prove regularity: bootstrap procedures using linear elliptic estimates(W 2,p, Lp-Lq, Lp

δ-Lqδ, . . . ). This can be combined with test-function, cut-off

or truncation arguments [Propositions 3.3, 3.5, and see M4.2 below]M3.2. To establish pointwise singularity estimates

(a) Integral estimates from M8.3 below, combined with a bootstrap proce-dure [Theorems 4.1 (psg < p < pS), 8.7]

(b) Rescaling, Liouville-type results and doubling arguments [Remark8.8(i)]

(c) Combination of the following three ingredients: the characterization ofnonnegative distributions with point support (the nonnegativity be-ing obtained by truncation and test-function techniques); a comparisonargument involving the Newton potential; a bootstrap procedure [The-orems 4.2, 4.1 (1 < p < psg)]

M3.3. To produce singular solutions(a) Method based on the construction and pointwise estimates of a singular

solution of the linear Laplace equation [Theorems 11.5, 31.16](b) Explicit singular solutions [Remarks 3.6(ii), 31.19, Formula (40.20)](c) ODE methods [Remark 3.6(ii)]

M4. Methods to prove a priori estimates

M4.1. Method of Hardy-Sobolev inequalities. Also: variant based on the use of asingular test-function [Section 10, Remark 31.18(i)]

M4.2. Bootstrap in Lpδ-spaces. Alternate bootstrap in the case of systems [Sec-

tion 11, Subsection 31.4]M4.3. Method based on rescaling and Liouville-type theorems [Section 12, Sub-

section 31.3]M4.4. Method of moving planes and Pohozaev-type identities [Section 10, Theo-

rem 31.2]

II. METHODS FOR PARABOLIC PROBLEMS

M5. Methods for local well-posedness

M5.1. Local existence-uniquenessNote: we are mainly concerned with irregular initial data. The case ofsmooth data is standard.(a) Fixed-point in a metric space of functions of t, with a weight vanishing

at t = 0, using Lp-Lq-estimates for the heat semigroup (variants: Lpδ-

spaces or uniformly local spaces, instead of Lp) [Theorems 15.2, 15.9,32.1(i), Remark 43.14(b)]

536 Appendices

(b) Similar to M5.1(a), with Lp-spaces replaced by a scale of interpolation-extrapolation spaces [Theorem 51.25]

(c) Improvement of the uniqueness class (without temporal weight): oneshows that any solution actually belongs to the fixed-point space; thisis achieved by a time-shift and continuous dependence argument [sameas M5.1(a)-M5.1(b)]

M5.2. Local nonexistence of positive solutions: for suitable singular initial data,contradiction between two pointwise estimates for the “free” part e−tAu0

(namely: a lower estimate for the linear heat equation as t → 0, and anupper a priori estimate depending on the nonlinear equation; see M8.5 fora closely related argument and more details) [Theorems 15.3, 15.10, 32.1(ii)]

M5.3. Local nonuniqueness(a) Nonuniqueness for zero initial data: construction of a forward self-

similar solution with exponential decay in space by ODE (shooting)methods [Remarks 15.4(ii), 40.11(a)]

(b) Construction of a singular stationary solution (which coexists with aclassical solution for t > 0) [Remark 15.4(iii)]

(c) Nonuniqueness for general initial data: method based on concentratedperturbations of an initial data, continuity of the existence time anduniversal bounds [Proposition 28.1]

M5.4. Regularity and smoothing: bootstrap procedure using (e.g.) Lp-Lq-esti-mates for the heat semigroup [Theorems 15.2, 15.9, 15.11, 43.13].

M5.5. Continuation properties (in particular: uniform bounds from Lq-bounds)(a) Consequence of well-posedness in M5.1(a)–M5.1(b) and smoothing

property in M5.4. Also, a lower estimate of the norm of u(·, t) near theblow-up time can be directly deduced from the fixed-point argument[Remark 16.2(iii), Theorem 33.5]

(b) Moser-type iteration [Theorems 16.4, 33.5, Remark 33.6](c) Variation-of-constants formula combined with interpolation inequality

and interpolation-extrapolation spaces [Proposition 51.34] (or just Lp-spaces [Theorem 32.2] )Note: in the case of systems, better results can be obtained by alternateuse of each equation

(d) Consequence of lower estimates on the blow-up profile (cf. M12.3(a))Note: non-continuation can be shown as a consequence of upper es-timates on the blow-up profile [Corollary 24.2, Theorem 44.6, Re-mark 44.8(c)]

(e) Energy arguments [Proposition 16.3](f) Gradient bounds (in particular via Bernstein techniques): see details in

[Section 35]


M6. Methods to prove global existence (and also boundedness, decay,stability)

M6.1. Multiplier, energy and Lyapunov functional methodsNote: the following three items partially overlap(a) Use of powers of the solution as multiplier, in combination with vari-

ous functional inequalities [Theorems 19.3(i), 33.9(i), 36.4(i), Remark40.11(b), Theorems 43.1, 44.5(i)]

(b) Potential well method [Theorem 19.5(i)](c) Use of a Lyapunov functional [Theorems 33.5, 33.18(ii), 40.7(i)]

M6.2. Comparison methods(a) Supersolutions with separated variables; spatially homogeneous super-

solutions [Theorems 19.2, 32.5(ii), 43.1, 46.1(ii)](b) Stationary supersolutions (and families thereof); singular steady-

states and their perturbations; quasi-stationary supersolutions [The-orem 19.15(ii), Remark 19.14, Theorems 20.5, 29.1 32.5(iii), 36.1(ii),36.4(i), 37.2, 40.7(iii), 44.17(i)]

(c) Supersolutions involving the heat semigroup (possibly self-similar)[Theorem 20.2, Remark 20.4(i), Theorems 20.6, 20.11, 32.5(ii) and37.4(ii)]

(d) Self-similar supersolutions [Theorem 20.6, Section 45](e) Traveling wave supersolutions [Theorem 36.7](f) Intersection-comparison with radial steady-states (to show global exis-

tence of a threshold solution) [Theorem 22.9]M6.3. Variation-of-constants formula and semigroup estimates (e.g., Lp-Lq or ex-

ponential decay) [Remark 19.4(b), Theorems 20.15, 33.1, 40.7(i), 40.10(i),Subsection 51.3]Note: sometimes combined with fixed-point arguments in spaces with tem-poral weight [Theorem 20.19, Corollary 20.20]

M6.4. Forward similarity variables (combined with construction of stable mani-folds) [Proposition 20.13]

M6.5. Duality method [Theorem 33.2, Remarks 33.4, 33.6, 33.15]M6.6. Gradient estimates [Proposition 40.5, Theorem 40.7(iii), Remark 40.11(d)]

M7. Methods to prove blow-up (in finite -or sometimes infinite- time)

Note 1: the methods in M7.1–M7.2 lead to a differential inequality for some func-tional of u(·, t).Note 2: the methods which are mainly motivated by (blow-up) results of Fujita-type do not appear here (see M8.4(a), M8.5, M8.6).

538 Appendices

M7.1. Eigenfunction method [Theorems 17.1, 17.3, 32.5(i), 33.16, 36.1(i), Remarks40.4(i) and (ii), Theorem 46.1]Note: other test-functions independent of u can sometimes be used [cf. The-orem 43.1(i)]

M7.2. Energy and multiplier methods(a) Energy and Holder’s inequality (in bounded domains) [Theorems 17.6,

44.14](b) Energy and concavity argument (in general domains) [Theorem 17.6](c) Potential well method [Theorem 19.5(ii)](d) Use of a power of the solution as test-function, in combination with

various functional inequalities [Theorems 40.2, 41.1]M7.3. Comparison methods

(a) Blowing-up self-similar subsolutions [Theorems 36.2, Section 45](b) Other forms of subsolutions (perturbation of singular or regular steady

states, expanding waves, traveling waves, quasi-stationary, . . . ) [Theo-rems 29.1, 36.4(ii), Lemma 36.6, Theorem 41.1, Remark 40.4(i), Theo-rem 44.17(ii)]

(c) Blow-up above a positive equilibrium [Theorem 17.8, Proposition 19.11](d) Comparison between domains [Remark 17.14](e) Cf. M11.2(a) [Theorem 23.5]

Note: in spatially nonlocal problems, this is sometimes combined withthe method in M12.1(a) to obtain preliminary estimates on the nonlocalterm [Theorem 44.5(ii)]

M7.4. Use of scaling properties of the equation (e.g. to prove blow-up for initialdata with slow decay at infinity)(a) Rescaled eigenfunctions [Theorem 17.12](b) Rescaled subsolutions [Remarks 17.13(i), 36.3(iii), Theorems 19.3(ii),

36.4(ii)]M7.5. Construction of explicit blowing-up solutions (often under self-similar form,

or by solving an ODE) [Theorems 33.9(ii), 33.12, 33.18(i)]M7.6. Use of dynamical systems arguments (ω-limits via a strict Lyapunov func-

tional, or via monotonicity) combined with the absence of steady-states(may lead to blow-up in finite or infinite time) [Remark 19.14, Theo-rems 28.7(iv), 33.14]

M7.7. Direct estimation via integration in space-time parabolas and Sobolev in-equality (to prove growth of mass for a Cauchy problem) [Theorem 40.10(ii)]

M8. Methods to prove nonexistence in Liouville and Fujita-type results

Note: the methods in M8.1–M8.3 concern both elliptic and parabolic problems


M8.1. Rescaled test-functions(a) Spatial test-functions

[Theorems 8.4, 31.12] in the elliptic case; [Remark 18.2(i), Theorems32.7, 37.4] in the parabolic case (where this leads to differential in-equalities)

(b) Space-time test-functions [Theorems 18.1(i), 37.1]M8.2. Moving plane methods

(a) Via symmetry, using the Kelvin transform (elliptic; case of the wholespace) [Theorem 8.1]

(b) Via symmetry and reduction to a one-dimensional problem on a half-line (elliptic; case of a half-space) [Theorem 8.2]

(c) Via monotonicity and reduction to an (n − 1)-dimensional problem inthe whole space (elliptic and parabolic; case of a half-space) [Theorems8.3, 21.8, 31.10]

M8.3. Integral estimates, obtained by using Bochner’s identity, power changeof dependent variable, and multipliers involving powers of u and cut-offs[Propositions 8.6, 21.5]

M8.4. Comparison methods(a) Families of blowing-up self-similar subsolutions [Section 45](b) Intersection-comparison with radial steady-states [Theorem 21.1]

M8.5. Method based on the variation-of-constants formula (for Fujita-type results)[Theorem 18.3]More precisely, a contradiction is obtained by comparing two pointwiseestimates for the “free” part e−tAu0 of the solution: the lower estimate fromthe linear heat equation as t→∞, and an upper a priori estimate dependingon the nonlinear equation [Lemma 15.6]. The latter is proved by taking theaction of the heat semigroup on the variation-of-constants formula.9 In thecritical case, the necessary additional information is provided by an L1 lowerbound based on convolution properties of Gaussians.

M8.6. Forward similarity variables [Lemma 18.4]

M9. Methods to prove boundedness of global solutions and parabolica priori estimates

Note: the methods in M9.1(b), M9.2(b), M9.3(a) here yield only boundedness ofglobal solutions

9For Fujita-type problems, the methods in M8.1(a) and M8.5 are essentially equivalent.In fact, in M8.1(a), one also compares a lower asymptotic estimate with an upper a priori bound.The latter follows from differential inequalities obtained by multiplying with rescaled Gaussiantest-functions, and these Gaussians are nothing but the heat kernel with time as a parameter.However the argument in M8.1(a) requires more regularity on the solution. Alternatively, theupper a priori bound can be obtained by a subsolution argument (see Remark 15.7).

540 Appendices

M9.1. Rescaling methods(a) Method based on rescaling, elliptic Liouville-type theorems and energy

[Theorem 22.1](b) Method based on rescaling and intersection-comparison, using the in-

finite intersection property of the singular and regular steady-states inR

n [Remark 23.13](c) Cf. M10.2(a) [Theorem 38.1]

M9.2. Energy methods(a) Method based on energy estimates and on a bootstrap argument using

interpolation and maximal regularity [Theorem 22.1, Proposition 22.11,Remark 44.15]

(b) Method based on energy estimates in forward similarity variables, andon a measure argument [Lemma 18.4]

M9.3. Methods based on the maximum principle(a) Intersection-comparison with a backward self-similar solution and with

the singular steady-state [Theorem 22.4](b) Use of a monotonicity property: the solution becomes increasing in time

if it reaches a sufficiently high level (for a nonlocal problem) [Proposi-tion 43.16]

M10. Methods to prove universal bounds of positive solutions andinitial blow-up rates

M10.1. Methods based on smoothing estimates(a) Smoothing in Lp

δ-spaces (using integral bounds obtained by the eigen-function method, and possibly combined with a priori estimates) [The-orems 26.1, 26.14]

(b) Smoothing in Lp-spaces (using integral bounds obtained by using asingular test-function, or by the eigenfunction method, and combinedwith a priori estimates) [Theorems 26.1, 43.15]

(c) Smoothing in uniformly local Lebesgue spaces (using integral boundsobtained by the eigenfunction method) [Theorem 26.13]

M10.2. Rescaling methods(a) Method based on a doubling lemma and parabolic Liouville-type the-

orems [Theorems 26.8, 26.9, 38.1](b) Method based on energy, measure arguments, and elliptic Liouville-

type theorems. [Theorem 26.6, Remark 26.7]M10.3. Methods based on space-time integral estimates

(a) Via Moser-type iteration or Harnack inequality [Theorem 26.13(i)](b) Via the method in M8.3 combined with Harnack inequality [Theo-

rem 26.8]


M11. Methods to establish blow-up rates

M11.1. Lower estimates(a) Comparison with solutions of the ODE [Proposition 23.1, Remark

38.2(i)](b) Differential inequality obtained by considering points of maxima of

u(·, t) [Proposition 23.1, Theorems 32.9, 44.2(i), 44.17(ii), 46.4(i),44.2(i), Proposition 44.3(i), Theorem 44.17(ii)]

(c) Variation-of-constants formula and use of the doubling time of ‖u(t)‖∞[Remark 23.2(ii)]

(d) Regularity estimates applied to the equation for ut (for the GBU prob-lem) [Theorem 40.18]

(e) Method using the intersections of the solution with a steady-state, andthe boundedness of ut (for the GBU problem) [Theorem 40.19]

M11.2. Upper estimates10

(a) Maximum principle applied to an auxiliary function J (for time-increasing solutions) [Theorems 23.5, 32.9, Remark 38.2(ii), Theo-rem 46.4]. See also [Theorem 40.21] for a different type of auxiliaryfunction in the GBU problem.

(b) Methods based on backward similarity variables(b)-1 Via rescaling, elliptic Liouville-type theorems and energy [The-

orem 23.7](b)-2 Via localized energy estimates and bootstrap [Remark 23.14(i)]

Note: the last two methods are similar to M9.1(a) and M9.2(a),respectively, the question being equivalent to the boundednessof global solutions for the equation in similarity variables

(c) Method based on rescaling and intersection-comparison, using the in-finite intersection property of the singular and regular steady-states inR

n [Theorem 23.10](d) Cf. M10.2(a) [Theorems 26.8, 38.1; see also Remarks 26.12, 32.12(i),

Proposition 44.3(ii)](e) Cf. M10.3(a) [Remark 32.12(i)](f) Cf. M12.4 [Subsection 43.2, Theorem 44.2(i)]

M12. Methods to study blow-up sets and profiles

M12.1. Methods based on the maximum principle(a) Maximum principle applied to an auxiliary function J (to obtain

single-point blow-up and upper profile estimates for radial nonin-

10One could alternatively classify the methods for upper blow-up estimates between:those using scaling and energy (M11.2(b)), those using scaling without energy (M11.2(c)–M11.2(d)), and those using neither energy nor scaling (M10.3(a), M11.2(a), M12.4).

542 Appendices

creasing solutions) [Theorem 24.1, Remark 32.12(ii), Theorems 39.7,44.2(iii), 44.6 (ii), Remark 46.5]Note: sometimes combined with a bootstrap argument [cf. Theo-rem 39.1]

(b) Moving plane method (to prove compactness of the blow-up set) [Re-mark 24.6(iv)]

(c) Sub-/supersolutions of blowing-up barrier type, using the notion ofsub-/super-standard functions (to obtain the blow-up behavior in theboundary layer for spatially nonlocal problems) [Theorem 43.10]

(d) Bernstein-type techniques (for blow-up profile estimates in the GBUproblem) [Remark 40.17, 41.4]

M12.2. Method of backward similarity variables(a) Combined with weighted energy and dynamical systems arguments (to

show asymptotically self-similar blow-up behavior; to exclude blow-upat a given point and prove compactness of the blow-up set) [Theo-rems 25.1, 24.5]Note: sometimes combined with comparison and cut-off arguments[Theorem 25.3]

(b) Combined with linearization and spectral techniques (to obtain refinedblow-up estimates and classification of profiles) [Remark 25.8]

(c) Construction of exact backward self-similar solutions by ODE (phaseplane) methods [Proposition 22.5, Remarks 39.8(i), (iii) and (iv)]

M12.3. Methods based on ODE’s in space(a) ODE energy estimate and use of the point of half-maximum of u(·, t)

(to obtain lower profile estimates for radial nonincreasing solutions)[Theorems 24.3, 39.2, Remark 32.12(ii)]

(b) Differential inequalities in space, relying on the boundedness of thetime derivative (for blow-up profile estimates in the GBU problem)[Theorem 40.14]

M12.4. Method based on eigenfunction arguments, one-sided estimates of ∆u (viathe maximum principle), and the mean-value inequality for subharmonicfunctions (to obtain the blow-up rate, set and profile for spatially nonlocalproblems) [Subsection 43.2, Theorem 44.2(i)]

Bibliography

[1] M. Abramowitz and I. Stegun, Handbook of mathematical functions withformulas, graphs, and mathematical tables,, Wiley, New York, 1972.

[2] N. Ackermann and T. Bartsch, Superstable manifolds of semilinear parabolicproblems, J. Dynam. Differential Equations 17 (2005), 115-173.

[3] N. Ackermann, T. Bartsch, P. Kaplicky and P. Quittner, A priori bounds,nodal equilibria and connecting orbits in indefinite superlinear parabolic prob-lems, Trans. Amer. Math. Soc. (to appear).

[4] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand,Princeton, N.J., 1965.

[5] J. Aguirre and M. Escobedo, On the blow up of solutions for a convectivereaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993),449-470.

[6] N. Alaa, Solutions faibles d’equations paraboliques quasi-lineaires avec don-nees initiales mesures, Ann. Math. Blaise Pascal 3 (1996), 1-15.

[7] S. Alama and M. Del Pino, Solutions of elliptic equations with indefinitenonlinearities via Morse theory and linking, Ann. Inst. H. Poincare Anal.Non Lineaire 13 (1996), 95-115.

[8] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinitenonlinearities, Calc. Var. Partial Differential Equations 1 (1993), 439-475.

[9] F. Alessio, P. Caldiroli and P. Montecchiari, Infinitely many solutions for aclass of semilinear elliptic equations in R

N , Boll. Unione Mat. Ital. (8) 4-B(2001), 311-318.

[10] L. Alfonsi and F.B. Weissler, Blow up in Rn for a parabolic equation with a

damping nonlinear gradient term, Progress in nonlinear differential equations(N.G. Lloyd et al., eds.), Birkhauser, Basel, 1992.

[11] N.D. Alikakos, Lp bounds of solutions of reaction-diffusion equations, Comm.Partial Differential Equations 4 (1979), 827-868.

[12] N.D. Alikakos, An application of the invariance principle to reaction diffusionequations, J. Differential Equations 33 (1979), 201-225.

[13] H. Amann, Existence and regularity for semilinear parabolic evolution equa-tions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 593-676.

[14] H. Amann, Parabolic evolution equations and nonlinear boundary conditions,J. Differential Equations 72 (1988), 201-269.

544 Bibliography

[15] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolicboundary value problems, Function Spaces, Differential Operators and Non-linear Analysis (H.J. Schmeisser and H. Triebel, eds.), Teubner, Stuttgart,Leipzig, 1993, pp. 9-126.

[16] H. Amann, Linear and quasilinear parabolic problems, Volume I: Abstractlinear theory, Birkhauser, Basel, 1995.

[17] H. Amann, Linear and quasilinear parabolic problems, Volume II, in prepa-ration.

[18] H. Amann, M. Hieber and G. Simonett, Bounded H∞-calculus for ellipticoperators, Differential Integral Equations 7 (1994), 613-653.

[19] H. Amann and J. Lopez-Gomez, A priori bounds and multiple solutions forsuperlinear indefinite elliptic problems, J. Differential Equations 146 (1998),336-374.

[20] H. Amann and P. Quittner, Elliptic boundary value problems involving mea-sures: existence, regularity, and multiplicity, Adv. Differential Equations 3(1998), 753-813.

[21] H. Amann and P. Quittner, Semilinear parabolic equations involving mea-sures and low regularity data, Trans. Amer. Math. Soc. 356 (2004), 1045-1119.

[22] H. Amann and P. Quittner, Optimal control problems with final observa-tion governed by explosive parabolic equations, SIAM J. Control Optim. 44(2005), 1215-1238.

[23] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in criticalpoint theory and applications, J. Funct. Anal. 14 (1973), 349-381.

[24] A. Ambrosetti and P.N. Srikanth, Superlinear elliptic problems and the dualprinciple in critical point theory, J. Math. Phys. Sci. 18 (1984), 441-451.

[25] A. Ambrosetti and M. Struwe, A note on the problem −∆u = λu+u|u|2∗−2,Manuscripta Math. 54 (1986), 373-379.

[26] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamil-ton-Jacobi equations, Nonlinear Anal. 31 (1998), 621-628.

[27] D. Andreucci, Degenerate parabolic equations with initial data measures,Trans. Amer. Math. Soc. 349 (1997), 3911-3923.

[28] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial tracesfor a class of evolution equations with strongly nonlinear sources, Ann. ScuolaNorm. Sup. Pisa Cl. Sci. (4) 18 (1991), 363-441.

[29] D. Andreucci, M.A. Herrero and J.J.L. Velazquez, Liouville theorems andblow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H.Poincare Anal. Non Lineaire 14 (1997), 1-53.

[30] S. Angenent, The zero set of a solution of a parabolic equation, J. ReineAngew. Math. 390 (1988), 79-96.

Bibliography 545

[31] S. Angenent and M. Fila, Interior gradient blow-up in a semilinear parabolicequation, Differential Integral Equations 9 (1996), 865-877.

[32] S. Angenent and R. van der Vorst, A priori bounds and renormalized Morseindices of solutions of an elliptic system, Ann. Inst. H. Poincare Anal. NonLineaire 17 (2000), 277-306.

[33] S.N. Antontsev and M. Chipot, The thermistor problem: existence, smooth-ness, uniqueness, blow-up, SIAM J. Math. Anal. 25 (1994), 1128-1156.

[34] G. Arioli, F. Gazzola, H.-Ch. Grunau and E. Sassone, The second bifurcationbranch for radial solutions of the Brezis-Nirenberg problem in dimension four,NoDEA Nonlinear Differential Equations Appl. (to appear).

[35] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusionsarising in population genetics, Adv. Math. 30 (1978), 33-76.

[36] J.M. Arrieta and A. Rodrıguez-Bernal, Non well posedness of parabolic equa-tions with supercritical nonlinearities, Comm. Contemp. Math. 6 (2004),733-764.

[37] J.M. Arrieta, A. Rodrıguez-Bernal, J.W. Cholewa and T. Dlotko, Linearparabolic equations in locally uniform spaces, Math. Models Methods Appl.Sci. 14 (2004), 253-293.

[38] J.M. Arrieta, A. Rodrıguez-Bernal and Ph. Souplet, Boundedness of globalsolutions for nonlinear parabolic equations involving gradient blow-up phe-nomena, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 1-15.

[39] F.V. Atkinson, H. Brezis and L.A. Peletier, Nodal solutions of elliptic equa-tions with critical Sobolev exponents, J. Differential Equations 85 (1990),151-170.

[40] F.V. Atkinson and L.A. Peletier, Emden-Fowler equations involving criticalexponents, Nonlinear Anal. 10 (1986), 755-776.

[41] F.V. Atkinson and L.A. Peletier, Large solutions of elliptic equations involv-ing critical exponents, Asymptotic Anal. 1 (1988), 139-160.

[42] F.V. Atkinson and L.A. Peletier, Oscillations of solutions of perturbed au-tonomous equations with an application to nonlinear elliptic eigenvalue prob-lems involving critical Sobolev exponents, Differential Integral Equations 3(1990), 401-433.

[43] T. Aubin, Problemes isoperimetriques de Sobolev, J. Differential Geometry11 (1976), 573-598.

[44] P. Aviles, On isolated singularities in some nonlinear partial differentialequations, Indiana Univ. Math. J. 32 (1983), 773-791.

[45] A. Bahri, Topological results on a certain class of functionals and application,J. Funct. Anal. 41 (1981), 397-427.

[46] A. Bahri and H. Berestycki, A perturbation method in critical point theoryand applications, Trans. Amer. Math. Soc. 267 (1981), 1-32.

546 Bibliography

[47] A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving thecritical Sobolev exponent: the effect of the topology of the domain, Comm.Pure Appl. Math. 41 (1988), 253-294.

[48] A. Bahri and P.-L. Lions, Morse index of some min-max critical points.I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988),1027-1037.

[49] A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and theirMorse indices, Comm. Pure Appl. Math. 45 (1992), 1205-1215.

[50] A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinearelliptic equations in unbounded domains, Ann. Inst. H. Poincare Anal. NonLineaire 14 (1997), 365-413.

[51] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolu-tion equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), 473-486.

[52] A. Barabanova, On the global existence of solutions of a reaction-diffusionequation with exponential nonlinearity, Proc. Amer. Math. Soc. 122 (1994),827-831.

[53] P. Baras, Non unicite des solutions d’une equation d’evolution non lineaire,Ann. Fac. Sci. Toulouse Math. (5) 5 (1983), 287-302.

[54] P. Baras and L. Cohen, Complete blow-up after Tmax for the solution of asemilinear heat equation, J. Funct. Anal. 71 (1987), 142-174.

[55] P. Baras and R. Kersner, Local and global solvability of a class of semilinearparabolic equations, J. Differential Equations 68 (1987), 238-252.

[56] P. Baras and M. Pierre, Critere d’existence de solutions positives pour desequations semi-lineaires non monotones, Ann. Inst. H. Poincare Anal. NonLineaire 2 (1985), 185-212.

[57] G. Barles and F. Da Lio, On the generalized Dirichlet problem for viscousHamilton-Jacobi equations, J. Math. Pures Appl. 83 (2004), 53-75.

[58] J.-Ph. Bartier, Global behavior of solutions of a reaction diffusion equationwith gradient absorption in unbounded domains, Asympt. Anal. 46 (2006),325-347.

[59] J.-Ph. Bartier and Ph. Souplet, Gradient bounds for solutions of semilinearparabolic equations without Bernstein’s quadratic condition, C. R. Acad. Sci.Paris Ser. I Math. 338 (2004), 533-538.

[60] J. Bebernes, A. Bressan and A.A. Lacey, Total blow-up versus single-pointblow-up, J. Differential Equations 73 (1988), 30-44.

[61] J. Bebernes and S. Bricher, Final time blowup profiles for semilinear para-bolic equations via center manifold theory, SIAM J. Math. Anal. 23 (1992),852-869.

[62] J. Bebernes and D. Eberly, A description of self-similar blow-up for dimen-sions n ≥ 3, Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988), 1-21.

Bibliography 547

[63] J. Bebernes and D. Eberly, Mathematical problems from combustion theory,Springer, New York, 1989.

[64] J. Bebernes and A.A. Lacey, Finite-time blowup for a particular parabolicsystem, SIAM J. Math. Anal. 21 (1990), 1415-1425.

[65] J. Bebernes and A.A. Lacey, Finite time blowup for semilinear reactive-diffusive systems, J. Differential Equations 95 (1992), 105-129.

[66] J. Bebernes and A.A. Lacey, Global existence and finite-time blow-up for aclass of nonlocal parabolic problems, Adv. Differential Equations 2 (1996),927-953.

[67] H. Bellout, Blow-up of solutions of parabolic equations with nonlinear mem-ory, J. Differential Equations 70 (1987), 42-68.

[68] V. Benci and G. Cerami, The effect of the domain topology on the number ofpositive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal.114 (1991), 79-93.

[69] S. Benachour, S. Dabuleanu-Hapca and Ph. Laurencot, Decay estimates fora viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundaryconditions, Asympt. Anal. 51 (2007), 209-229.

[70] S. Benachour, G. Karch and Ph. Laurencot, Asymptotic profiles of solutionsto viscous Hamilton-Jacobi equations, J. Math. Pures Appl. 83 (2004), 1275-1308.

[71] S. Benachour and Ph. Laurencot, Global solutions to viscous Hamilton-Jacobi equations with irregular data, Comm. Partial Differential Equations24 (1999), 1999-2021.

[72] M. Ben-Artzi, Ph. Souplet and F.B. Weissler, The local theory for viscousHamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl. 81(2002), 343-378.

[73] R.D. Benguria, J. Dolbeault and M.J. Esteban, Classification of the solutionsof semilinear elliptic problems in a ball, J. Differential Equations 167 (2000),438-466.

[74] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefiniteelliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlin-ear Anal. 4 (1994), 59-78.

[75] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methodsfor indefinite superlinear homogeneous elliptic problems, NoDEA NonlinearDifferential Equations Appl. 2 (1995), 553-572.

[76] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existenceof a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-345.

[77] J. Bergh and J. Lofstrom, Interpolation spaces. An introduction, Springer,Berlin - Heidelberg - New York, 1976.

548 Bibliography

[78] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equa-tions on R

n or Rn+ through the method of moving planes, Comm. Partial

Differential Equations 22 (1997), 1671-1690.

[79] M.-F. Bidaut-Veron, Initial blow-up for the solutions of a semilinear para-bolic equation with source term, Equations aux derivees partielles et appli-cations, articles dedies a Jacques-Louis Lions, Gauthier-Villars, Paris, 1998,pp. 189-198.

[80] M.-F. Bidaut-Veron, Local behaviour of the solutions of a class of nonlinearelliptic systems, Adv. Differential Equations 5 (2000), 147-192.

[81] M.-F. Bidaut-Veron, A.C. Ponce and L. Veron, Boundary singularities ofpositive solutions of some nonlinear elliptic equations, C. R. Math. Acad.Sci. Paris 344 (2007), 83-88.

[82] M.-F. Bidaut-Veron and Th. Raoux, Asymptotics of solutions of some non-linear elliptic systems, Comm. Partial Differential Equations 21 (1996), 1035-1086.

[83] M.-F. Bidaut-Veron and L. Veron, Nonlinear elliptic equations on compactRiemannian manifolds and asymptotics of Emden equations, Invent. Math.106 (1991), 489-539.

[84] M.-F. Bidaut-Veron and L. Vivier, An elliptic semilinear equation withsource term involving boundary measures: the subcritical case, Rev. Mat.Iberoamericana 16 (2000), 477-513.

[85] M.-F. Bidaut-Veron and C. Yarur, Semilinear elliptic equations and systemswith measure data: existence and a priori estimates, Adv. Differential Equa-tions 7 (2002), 257-296.

[86] P. Biler, M. Guedda and G. Karch, Asymptotic properties of solutions of theviscous Hamilton-Jacobi equation, J. Evol. Equ. 4 (2004), 75-97.

[87] I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities andapplications, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1217-1247.

[88] L. Boccardo, F. Murat and J.-P. Puel, Existence results for some quasilinearparabolic equations, Nonlinear Anal. 13 (1989), 373-392.

[89] M. Bouhar and L. Veron, Integral representation of solutions of semilinearelliptic equations in cylinders and applications, Nonlinear Anal. 23 (1994),275-296.

[90] H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983.

[91] H. Brezis, unpublished manuscript.

[92] H. Brezis and X. Cabre, Some simple nonlinear PDE’s without solutions,Boll. Unione Mat. Ital. (8) 1-B (1999), 223-262.

[93] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initialdata, J. Anal. Math. 68 (1996), 277-304.

Bibliography 549

[94] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for ut −∆u = g(u) revisited, Adv. Differential Equations 1 (1996), 73-90.

[95] H. Brezis and A. Friedman, A nonlinear parabolic equations involving mea-sures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97.

[96] H. Brezis and T. Kato, Remarks on the Schrodinger operator with singularcomplex potentials, J. Math. Pures Appl. 58 (1979), 137-151.

[97] H. Brezis and P.-L. Lions, A note on isolated singularities for linear ellipticequations, Mathematical analysis and applications, Part A, Adv. in Math.Suppl. Stud., 7a,, Academic Press, New York-London, 1981, pp. 263-266.

[98] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equationsinvolving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983),437-477.

[99] H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems,Comm. Partial Differential Equations 2 (1977), 601-614.

[100] J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heatequations, Nonlinearity 7 (1994), 539-575.

[101] P. Brunovsky and B. Fiedler, Number of zeros on invariant manifolds inreaction-diffusion equations, Nonlinear Anal. 10 (1986), 179-193.

[102] P. Brunovsky, P. Polacik and B. Sandstede, Convergence in general periodicparabolic equations in one space dimension, Nonlinear Anal. 18 (1992), 209-215.

[103] C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equationwith conserved first integral, SIAM J. Appl. Math. 53 (1993), 718-742.

[104] C. Budd and J. Norbury, Semilinear elliptic equations and supercriticalgrowth, J. Differential Equations 68 (1987), 169-197.

[105] C. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinearelliptic equation, J. Differential Equations 82 (1989), 207-218.

[106] J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden sys-tem, Indiana Univ. Math. J. 51 (2002), 37-51.

[107] X. Cabre and Y. Martel, Weak eigenfunctions for the linearization of ex-tremal elliptic problems, J. Funct. Anal. 156 (1998), 30-56.

[108] L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and localbehavior of semilinear elliptic equations with critical Sobolev growth, Comm.Pure Appl. Math. 42 (1989), 271-297.

[109] G. Cai, On the heat flow for the two-dimensional Gelfand equation, NonlinearAnal. (to appear).

[110] G. Caristi and E. Mitidieri, Blow-up estimates of positive solutions of aparabolic system, J. Differential Equations 113 (1994), 265-271.

550 Bibliography

[111] F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities:Sharp constants, existence (and nonexistence), and symmetry of extremalfunctions, Comm. Pure Appl. Math. 54 (2001), 229-258.

[112] T. Cazenave, F. Dickstein and F.B. Weissler, An equation whose Fujita crit-ical exponent is not given by scaling, Nonlinear Anal. (to appear).

[113] T. Cazenave and A. Haraux, Introduction aux problemes d’evolution semi-lineaires, Ellipses, Paris, 1990, English translation: The Clarendon Press,Oxford University Press, New York, 1998.

[114] T. Cazenave and P.-L. Lions, Solutions globales d’equations de la chaleursemi lineaires, Comm. Partial Differential Equations 9 (1984), 955-978.

[115] T. Cazenave and F.B. Weissler, Asymptotically self-similar global solutions ofthe nonlinear Schrodinger and heat equations, Math. Z. 228 (1998), 83-120.

[116] C. Celik and Z. Zhou, No local L1 solution for a nonlinear heat equation,Comm. Partial Differential Equations 28 (2003), 1807-1831.

[117] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinearelliptic boundary value problems involving critical exponents, J. Funct. Anal.69 (1986), 289-306.

[118] K. Cerqueti, A uniqueness result for a semilinear elliptic equation involvingthe critical Sobolev exponent in symmetric domains, Asymptot. Anal. 21(1999), 99-115.

[119] J.M. Chadam, A. Peirce and H.-M. Yin, The blow-up property of solutions tosome diffusion equations with localized nonlinear reactions, J. Math. Anal.Appl. 169 (1992), 313-328.

[120] C.-Y. Chan, Recent advances in quenching phenomena, Proceedings of Dy-namic Systems and Applications, Vol. 2, Dynamic, Atlanta, GA, 1996, pp.107-113.

[121] K.-C. Chang and M.-Y. Jiang, Dirichlet problem with indefinite nonlineari-ties, Calc. Var. Partial Differential Equations 20 (2004), 257-282.

[122] H. Chen, Positive steady-state solutions of a non-linear reaction-diffusionsystem, Math. Methods Appl. Sci. 20 (1997), 625-634.

[123] W. Chen and C. Li, Classification of solutions of some nonlinear ellipticequations, Duke Math. J. 63 (1991), 615-622.

[124] W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin.Dyn. Syst. 3 (1997), 333-340.

[125] X. Chen, M. Fila and J.-S. Guo, Boundedness of global solutions of a super-critical parabolic equation, Nonlinear Anal. (to appear).

[126] X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. DifferentialEquations 78 (1989), 160-190.

Bibliography 551

[127] X.-Y. Chen and P. Polacik, Asymptotic periodicity of positive solutions ofreaction diffusion equations on a ball, J. Reine Angew. Math. 472 (1996),17-51.

[128] M. Chipot and P. Quittner, Equilibria, connecting orbits and a priori boundsfor semilinear parabolic equations with nonlinear boundary conditions, J. Dy-nam. Differential Equations 16 (2004), 91-138.

[129] M. Chipot and F.B. Weissler, Some blow up results for a nonlinear parabolicproblem with a gradient term, SIAM J. Math. Anal. 20 (1989), 886-907.

[130] M. Chlebık and M. Fila, From critical exponents to blowup rates for parabolicproblems, Rend. Mat. Appl. (7) 19 (1999), 449-470.

[131] M. Chlebık, M. Fila and P. Quittner, Blow-up of positive solutions of asemilinear parabolic equation with a gradient term, Dyn. Contin. DiscreteImpuls. Syst. Ser. A Math. Anal. 10 (2003), 525-537.

[132] K.-S. Chou, S.-Z. Du and G.-F. Zheng, On partial regularity of the border-line solution of semilinear parabolic problems, Calc. Var. Partial DifferentialEquations (to appear).

[133] V. Churbanov, An example of a reaction system with diffusion in which thediffusion terms lead to blowup, Dokl. Akad. Nauk SSSR 310 (1990), 1308-1309, English translation in: Soviet Math. Dokl. 41 (1990), 191-192.

[134] Ph. Clement, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semi-linear elliptic systems, Comm. Partial Differential Equations 17 (1992), 923-940.

[135] Ph. Clement, D.G. de Figueiredo and E. Mitidieri, A priori estimates forpositive solutions of semilinear elliptic systems via Hardy-Sobolev inequali-ties, Nonlinear partial differential equations, Pitman Res. Notes Math. Ser.343 (A. Benkirane et al., eds.), Harlow: Longman, 1996, pp. 73-91.

[136] Ph. Clement, J. Fleckinger, E. Mitidieri and F. de Thelin, Existence of posi-tive solutions for nonvariational quasilinear system, J. Differential Equations166 (2000), 455-477.

[137] Ph. Clement and R.C.A.M. van der Vorst, On a semilinear elliptic system,Differential Integral Equations 8 (1995), 1317-1329..

[138] G. Conner and C. Grant, Asymptotics of blowup for a convection-diffusionequation with conservation, Differential Integral Equations 9 (1996), 719-728.

[139] M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equa-tions on R

N . Part I: A global variational approach, Arch. Rational Mech.Anal. 153 (2000), 291-316.

[140] M. Conti and S. Terracini, Radial solutions of superlinear equations on RN .

Part II: The forced case, Arch. Rational Mech. Anal. 153 (2000), 317-339.[141] C. Cosner, Positive solutions for a superlinear elliptic systems without vari-

ational structure, Nonlinear Anal. 8 (1984), 1427-1436.

552 Bibliography

[142] M.G. Crandall and P.H. Rabinowitz, Some continuation and variationalmethods for positive solutions of nonlinear elliptic eigenvalue problems, Arch.Rational Mech. Anal. 58 (1975), 207-218.

[143] M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem witha singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222.

[144] M. Cuesta, D.G. de Figueiredo and P.N. Srikanth, On a resonant-superlinearelliptic problem, Calc. Var. Partial Differential Equations 17 (2003), 221-233.

[145] S. Cui, Local and global existence of solutions to semilinear parabolic initialvalue problems, Nonlinear Anal. 43 (2001), 293-323.

[146] S. Dabuleanu, Problemes aux limites pour les equations de Hamilton-Jacobiavec viscosite et donnees initiales peu regulieres, Doctoral Thesis, Universityof Nancy 1, 2003.

[147] L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positivesolutions of semilinear elliptic equations in symmetric domains via the max-imum principle, Ann. Inst. H. Poincare Anal. Non Lineaire 16 (1999), 631-652.

[148] E.N. Dancer, The effect of domain shape on the number of positive solutionsof certain nonlinear equations, J. Differential Equations 74 (1988), 120-156.

[149] E.N. Dancer, A note on an equation with critical exponent, Bull. LondonMath. Soc. 20 (1988), 600-602.

[150] E.N. Dancer, Some notes on the method of moving planes, Bull. Austral.Math. Soc. 46 (1992), 425-434.

[151] E.N. Dancer, Superlinear problems on domains with holes of asymptoticshape and exterior problems, Math. Z. 229 (1998), 475-491.

[152] D. Daners and P. Koch Medina, Abstract evolution equations, periodic prob-lems and applications, Longman, Harlow, 1992.

[153] E.B. Davies, The equivalence of certain heat kernel and Green functionbounds, J. Funct. Anal. 71 (1987), 88-103.

[154] E.B. Davies, Heat kernels and spectral theory, Cambridge University Press,Cambridge, 1989.

[155] M. Del Pino, M. Musso and F. Pacard, Boundary singularities for weaksolutions of semilinear elliptic problems, J. Funct. Anal. (to appear).

[156] K. Deng, Stabilization of solutions of a nonlinear parabolic equation with agradient term, Math. Z. 216 (1994), 147-155.

[157] K. Deng, Nonlocal nonlinearity versus global blow-up, Math. Applicata 8(1995), 124-129.

[158] K. Deng, Blow-up rates for parabolic systems, Z. Angew. Math. Phys. 47(1996), 132-143.

Bibliography 553

[159] K. Deng and H.A. Levine, The role of critical exponents in blow-up theorems:The sequel, J. Math. Anal. Appl. 243 (2000), 85-126.

[160] W. Deng, Y. Li and C.-H. Xie, Semilinear reaction-diffusion systems withnonlocal sources, Math. Comput. Modelling 37 (2003), 937-943.

[161] R. Denk, M. Hieber and J. Pruss, R-boundedness, Fourier multipliers andproblems of elliptic and parabolic type, Mem. Amer. Math. Soc. 788, 2003.

[162] G. Devillanova and S. Solimini, Concentration estimates and multiple solu-tions to elliptic problems at critical growth, Adv. Differential Equations 7(2002), 1257-1280.

[163] C. Dohmen and M. Hirose, Structure of positive radial solutions to theHaraux-Weissler equation, Nonlinear Anal. 33 (1998), 51-69.

[164] B. Dold, V.A. Galaktionov, A.A. Lacey and J.L. Vazquez, Rate of approachto a singular steady state in quasilinear reaction-diffusion equations, Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 663-687.

[165] G.C. Dong, Nonlinear partial differential equations of second order, Amer.Math. Soc., Transl. Math. Monographs 95, Providence, RI, 1991.

[166] G. Dore and A. Venni, H∞ functional calculus for an elliptic operator on ahalf-space with general boundary conditions, Ann. Scuola Norm. Sup. PisaCl. Sci. (5) 1 (2002), 487-543.

[167] Y. Du and S. Li, Nonlinear Liouville theorems and a priori estimates for in-definite superlinear elliptic equations, Adv. Differential Equations 10 (2005),841-860.

[168] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposi-tion de fonctions, C. R. Acad. Sci. Paris Ser. I Math. 315 (1992), 693-698.

[169] X.T. Duong and G. Simonett, H∞-calculus for elliptic operators with non-smooth coefficients, Differential Integral Equations 10 (1997), 201-217.

[170] L. Dupaigne and A. Ponce, Singularities of positive supersolutions in ellipticPDEs, Selecta Math. (N.S.) 10 (2004), 341-358.

[171] M. Escobedo and M.A. Herrero, Boundedness and blow up for a semilinearreaction-diffusion system, J. Differential Equations 89 (1991), 176-202.

[172] M. Escobedo and M.A. Herrero, A uniqueness result for a semilinear reactiondiffusion system, Proc. Amer. Math. Soc. 112 (1991), 175-185.

[173] M. Escobedo and M.A. Herrero, A semilinear parabolic system in a boundeddomain, Ann. Mat. Pura Appl. (4) 165 (1993), 315-336.

[174] M. Escobedo and O. Kavian, Variational problems related to self-similarsolutions of the heat equation, Nonlinear Anal. 11 (1987), 1103-1133.

[175] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusionequations in R

N , J. Funct. Anal. 10 (1991), 119-161.

554 Bibliography

[176] M.J. Esteban, On periodic solutions of superlinear parabolic problems, Trans.Amer. Math. Soc. 293 (1986), 171-189.

[177] M.J. Esteban, A remark on the existence of positive periodic solutions ofsuperlinear parabolic problems, Proc. Amer. Math. Soc. 102 (1988), 131-136.

[178] A. Farina, Liouville-type results for solutions of −∆u = |u|p−1u on un-bounded domains of R

N , C. R. Math. Acad. Sci. Paris 341 (2005), 415-418.

[179] P.C. Fife, Mathematical aspects of reacting and diffusing systems, LectureNotes in Biomathematics 28, Springer-Verlag, Berlin - New York, 1979.

[180] D.G. de Figueiredo, Semilinear elliptic systems, Nonlinear Functional Anal-ysis and Applications to Differential Equations, World Sci. Publishing, RiverEdge, N.J., 1998, pp. 122-152.

[181] D.G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic sys-tems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 387-397.

[182] D.G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans.Amer. Math. Soc. 343 (1994), 99-116.

[183] D.G. de Figueiredo, P.-L. Lions and R.D. Nussbaum, A priori estimates andexistence of positive solutions of semilinear elliptic equations, J. Math. PuresAppl. 61 (1982), 41-63.

[184] D.G. de Figueiredo and J. Yang, A priori bounds for positive solutions ofa non-variational elliptic system, Comm. Partial Differential Equations 26(2001), 2305-2321.

[185] M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gra-dient term, Proc. Amer. Math. Soc. 111 (1991), 795-801.

[186] M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J.Differential Equations 98 (1992), 226-240.

[187] M. Fila, Boundedness of global solutions of nonlocal parabolic equations, Non-linear Anal. 30 (1997), 877-885.

[188] M. Fila, Blow-up of solutions of supercritical parabolic equations, Handbookof Differential Equations, Evolutionary equations, Vol. II (C.M. Dafermos etal., eds.), Elsevier/North-Holland, Amsterdam, 2005, pp. 105-158.

[189] M. Fila, J.R. King, M. Winkler and E. Yanagida, Optimal lower bound of thegrow-up rate for a supercritical parabolic equation, J. Differential Equations228 (2006), 339-356.

[190] M. Fila, J.R. King, M. Winkler and E. Yanagida, Grow-up rate of solutionsof a semilinear parabolic equation with a critical exponent, Adv. DifferentialEquations 12 (2007), 1-26.

[191] M. Fila, J.R. King, M. Winkler and E. Yanagida, Linear behaviour of solu-tions of a superlinear heat equation (2006), Preprint.

Bibliography 555

[192] M. Fila, H.A. Levine and Y. Uda, Fujita-type global existence–global non-existence theorem for a system of reaction diffusion equations with differingdiffusivities, Math. Methods Appl. Sci. 17 (1994), 807-835.

[193] M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinearparabolic equations, Differential Integral Equations 7 (1994), 811-821.

[194] M. Fila, H. Matano and P. Polacik, Immediate regularization after blow-up,SIAM J. Math. Anal. 37 (2005), 752-776.

[195] M. Fila and N. Mizoguchi, Multiple continuation beyond blow-up, DifferentialIntegral Equations 20 (2007), 671-680.

[196] M. Fila and H. Ninomiya, Reaction versus diffusion: blow-up induced andinhibited by diffusivity, Russian Math. Surveys 60 (2005), 1217-1235.

[197] M. Fila, H. Ninomiya and J.L. Vazquez, Dirichlet boundary conditions canprevent blow-up in reaction-diffusion equations and systems, Discrete Contin.Dyn. Syst. 14 (2006), 63-74.

[198] M. Fila and P. Polacik, Global solutions of a semilinear parabolic equation,Adv. Differential Equations 4 (1999), 163-196.

[199] M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic prob-lems on general domains, NoDEA Nonlinear Differential Equations Appl. 8(2001), 473-480.

[200] M. Fila, Ph. Souplet and F.B. Weissler, Linear and nonlinear heat equa-tions in Lp

δ spaces and universal bounds for global solutions, Math. Ann.320 (2001), 87-113.

[201] M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady-stateof a parabolic equation with gradient blow-up, Appl. Math. Letters 20 (2007),578-582.

[202] M. Fila and M. Winkler, Single-point blow-up on the boundary where the zeroDirichlet boundary condition is imposed, J. Eur. Math. Soc. (to appear).

[203] M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a super-critical semilinear diffusion equation, J. Differential Equations 205 (2004),365-389.

[204] M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a para-bolic equation with supercritical nonlinearity, Math. Ann. (to appear).

[205] M. Fila, M. Winkler and E. Yanagida, Convergence to self-similar solutionsin a semilinear parabolic equation (2007), Preprint.

[206] S. Filippas, M.A. Herrero and J.J.L. Velazquez, Fast blow-up mechanismsfor sign-changing solutions of a semilinear parabolic equation with criticalnonlinearity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000),2957-2982.

[207] S. Filippas and R. Kohn, Refined asymptotics for the blowup of ut−∆u = up,Comm. Pure Appl. Math. 45 (1992), 821-869.

556 Bibliography

[208] S. Filippas and W.-X. Liu, On the blowup of multidimensional semilinearheat equations, Ann. Inst. H. Poincare Anal. Non Lineaire 10 (1993), 313-344.

[209] S. Filippas and F. Merle, Compactness and single-point blowup of positive so-lutions on bounded domains, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997),47-65.

[210] A. Filippov, Conditions for the existence of a solution of a quasi-linear par-abolic equation (Russian), Dokl. Akad. Nauk SSSR 141 (1961), 568-570.

[211] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7(1937), 335-369.

[212] D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinearelliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect.A 105 (1987), 205-213.

[213] R.H. Fowler, Further studies of Emden’s and similar differential equations,Quart. J. Math. 2 (1931), 259-288.

[214] A. Friedman, Partial differential equations of parabolic type, Prentice Hall,1964.

[215] A. Friedman, Blow up of solutions of nonlinear parabolic equations, Nonlineardiffusion equations and their equilibrium states, I (W.-M. Ni et al., eds.),Springer, 1988, pp. 301-318.

[216] A. Friedman and Y. Giga, A single point blow-up for solutions of semilinearparabolic systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), 65-79.

[217] A. Friedman and M. Herrero, Extinction properties of semilinear heat equa-tions with strong absorption, J. Math. Anal. Appl. 124 (1987), 530-546.

[218] A. Friedman and A.A. Lacey, Blowup of solutions of semilinear parabolicequations, J. Math. Anal. Appl. 132 (1988), 171-186.

[219] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heatequations, Indiana Univ. Math. J. 34 (1985), 425-447.

[220] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut =∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 13 (1966), 109-124.

[221] I. Fukuda and R. Suzuki, Blow-up behavior for a nonlinear heat equation witha localized source in a ball, J. Differential Equations 218 (2005), 273-291.

[222] V.A. Galaktionov, Geometric Sturmian theory of nonlinear parabolic equa-tions and applications, Applied Mathematics and Nonlinear Science Series,3, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[223] V.A. Galaktionov and J.R. King, Composite structure of global unboundedsolutions of nonlinear heat equations with critical Sobolev exponents, J. Dif-ferential Equations 189 (2003), 199-233.

[224] V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, A parabolic systemof quasilinear equations, I, Differential Equations 19 (1983), 2133-2143.

Bibliography 557

[225] V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, Asymptotic stabil-ity of invariant solutions of nonlinear heat-conduction equation with sources,Differentsial’nye Uravneniya 20 (1984), 614-632, (English translation Differ-ential Equations 20 (1984), 461-476).

[226] V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, A parabolic systemof quasilinear equations, II, Differential Equations 21 (1985), 1544-1559.

[227] V.A. Galaktionov and H.A. Levine, A general approach to critical Fujitaexponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), 1005-1027.

[228] V.A. Galaktionov and S. Posashkov, The equation ut = uxx + uβ. Localiza-tion, asymptotic, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint no. 97 (1985).

[229] V.A. Galaktionov and S. Posashkov, Application of new comparison theo-rems to the investigation of unbounded solutions of nonlinear parabolic equa-tions (Russian), Differentsial’nye Uravneniya 22 (1986), 1165-1173, (Englishtranslation: Differential Equations 22 (1986), 809-815).

[230] V.A. Galaktionov and J.L. Vazquez, Regional blow up in a semilinear heatequation with convergence to a Hamilton-Jacobi equation, SIAM J. Math.Anal. 24 (1993), 1254-1276.

[231] V.A. Galaktionov and J.L. Vazquez, Blowup for quasilinear heat equationsdescribed by means of nonlinear Hamilton-Jacobi equations, J. DifferentialEquations 127 (1996), 1-40.

[232] V.A. Galaktionov and J.L. Vazquez, Continuation of blow-up solutions ofnonlinear heat equations in several space dimensions, Comm. Pure Appl.Math. 50 (1997), 1-67.

[233] Th. Gallouet, F. Mignot and J.-P. Puel, Quelques resultats sur le probleme−∆u = λeu, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 289-292.

[234] F. Gazzola and H.-Ch. Grunau, On the role of space dimension n = 2+2√

2in the semilinear Brezis-Nirenberg eigenvalue problem, Analysis 20 (2000),395-399.

[235] F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free bound-ary problems for quasilinear elliptic operators, Adv. Differential Equations5 (2000), 1-30.

[236] F. Gazzola and T. Weth, Finite time blow-up and global solutions for semi-linear parabolic equations with initial data at high energy level, DifferentialIntegral Equations 18 (2005), 961-990.

[237] J. Giacomoni, J. Prajapat and M. Ramaswamy, Positive solution branchfor elliptic problems with critical indefinite nonlinearity, Differential IntegralEquations 18 (2005), 721-764.

558 Bibliography

[238] B. Gidas, Symmetry properties and isolated singularities of positive solutionsof nonlinear elliptic equations, Nonlinear partial differential equations in en-gineering and applied science, Lecture Notes in Pure and Appl. Math. 54,Dekker, New York, 1980, pp. 255-273.

[239] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties viathe maximum principle, Comm. Math. Phys. 68 (1979), 209-243.

[240] B. Gidas and J. Spruck, Global and local behavior of positive solutions ofnonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598.

[241] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinearelliptic equations, Comm. Partial Differential Equations 6 (1981), 883-901.

[242] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Ky-bernetik 12 (1972), 30-39.

[243] Y. Giga, A bound for global solutions of semilinear heat equations, Comm.Math. Phys. 103 (1986), 415-421.

[244] Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinearheat equations, Comm. Pure Appl. Math. 38 (1985), 297-319.

[245] Y. Giga and R.V. Kohn, Characterizing blowup using similarity variables,Indiana Univ. Math. J. 36 (1987), 1-40.

[246] Y. Giga and R.V. Kohn, Nondegeneracy of blowup for semilinear heat equa-tions, Comm. Pure Appl. Math. 42 (1989), 845-884.

[247] Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equa-tion with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483-514.

[248] Y. Giga, S. Matsui and S. Sasayama, On blow-up rate for sign-changingsolutions in a convex domain, Math. Methods Appl. Sc. 27 (2004), 1771-1782.

[249] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heatequations, J. Math. Anal. Appl. 316 (2006), 538-555.

[250] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of sec-ond order, Springer, Berlin, 1998.

[251] B. Gilding, The Cauchy problem for ut = ∆u + |∇u|q, large-time behaviour,J. Math. Pures Appl. 84 (2005), 753-785.

[252] B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for ut = ∆u+|∇u|q, J. Math. Anal. Appl. 284 (2003), 733-755.

[253] J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complexGinzburg Landau equation. II. Contraction methods, Comm. Math. Phys.187 (1997), 45-79.

[254] P. Groisman and J. Rossi, Dependence of the blow-up time with respect toparameters and numerical approximations for a parabolic problem, Asympt.Anal. 37 (2004), 79-91.

Bibliography 559

[255] P. Groisman, J. Rossi and H. Zaag, On the dependence of the blow-up timewith respect to the initial data in a semilinear parabolic problem, Comm.Partial Differential Equations 28 (2003), 737-744.

[256] M. Grossi, A uniqueness result for a semilinear elliptic equation in symmetricdomains, Adv. Differential Equations 5 (2000), 193-212.

[257] M. Grossi, P. Magrone and M. Matzeu, Linking type solutions for ellipticequations with indefinite nonlinearities up to the critical growth, DiscreteContin. Dyn. Syst. 7 (2001), 703-718.

[258] Y. Gu and M.-X. Wang, Existence of positive stationary solutions and thresh-old results for a reaction-diffusion system, J. Differential Equations 130(1996), 177-291.

[259] M. Guedda and M. Kirane, Diffusion terms in systems of reaction diffusionequations can lead to blow up, J. Math. Anal. Appl. 218 (1998), 325-327.

[260] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positivesteady states of a semilinear heat equation in R

n, Comm. Pure Appl. Math.45 (1992), 1153-1181.

[261] C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation,J. Differential Equations 169 (2001), 588-613.

[262] C. Gui and X. Wang, Life span of solutions of the Cauchy problem for asemilinear heat equation, J. Differential Equations 115 (1995), 166-172.

[263] J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with anonlinear gradient source term, Discrete Contin. Dyn. Syst. (to appear).

[264] J.-S. Guo and Ph. Souplet, Fast rate of formation of dead-core for the heatequation with strong absorption and applications to fast blow-up, Math. Ann.331 (2005), 651-667.

[265] J.K. Hale and G. Raugel, Convergence in gradient-like systems with appli-cations to PDE, Z. Angew. Math. Phys. 43 (1992), 63-124.

[266] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of theFisher-KPP equation in R

N , Arch. Ration. Mech. Anal. 157 (2001), 91-163.[267] A. Haraux, Systemes dynamiques dissipatifs et applications, Recherches en

Mathematiques Appliquees 17, Masson, Paris, 1991.[268] A. Haraux and P. Polacik, Convergence to a positive equilibrium for some

nonlinear evolution equations in a ball, Acta Math. Univ. Comenian. (N.S.)61 (1992), 129-141.

[269] A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial valueproblem, Indiana Univ. Math. J. 31 (1982), 167-189.

[270] A. Haraux and A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations, Tohoku Math. J. (2) 40 (1988), 159-163.

[271] K. Hayakawa, On nonexistence of global solutions of some semilinear para-bolic differential equations, Proc. Japan Acad. 49 (1973), 503-505.

560 Bibliography

[272] E. Heinz, Uber die Eindeutigkeit beim Cauchyschen Anfangswertproblem ei-ner elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss.Gottingen IIa (1955), 1-12.

[273] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notesin Mathematics 840, Springer, Berlin - Heidelberg - New York, 1981.

[274] M.A. Herrero, A.A. Lacey and J.J.L. Velazquez, Global existence for reac-tion-diffusion systems modelling ignition, Arch. Rational Mech. Anal. 142(1998), 219-251.

[275] M.A. Herrero and J.J.L. Velazquez, Blow-up profiles in one-dimensional,semilinear parabolic problems, Comm. Partial Differential Equations 17(1992), 205-219.

[276] M.A. Herrero and J.J.L. Velazquez, Blow-up behaviour of one-dimensionalsemilinear parabolic equations, Ann. Inst. H. Poincare Anal. Non Lineaire10 (1993), 131-189.

[277] M.A. Herrero and J.J.L. Velazquez, Explosion de solutions d’equations para-boliques semilineaires supercritiques, C. R. Acad. Sci. Paris Ser. I Math. 319(1994), 141-145.

[278] M.A. Herrero and J.J.L. Velazquez, A blow up result for semilinear heatequations in the supercritical case (1994), Preprint.

[279] M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family ofnonlinear parabolic equations in general domain in R

N , Michigan Math. J.52 (2004), 375-389.

[280] N. Hirano and N. Mizoguchi, Positive unstable periodic solutions for super-linear parabolic equations, Proc. Amer. Math. Soc. 123 (1995), 1487-1495.

[281] S. Hollis, R. Martin and M. Pierre, Global existence and boundedness inreaction-diffusion systems, SIAM J. Math. Anal. 18 (1987), 744-761.

[282] S. Hollis and J. Morgan, Interior estimates for a class of reaction-diffusionsystems from L1 a priori estimates, J. Differential Equations 98 (1992), 260-276.

[283] B. Hu, Remarks on the blowup estimate for solutions of the heat equation witha nonlinear boundary condition, Differential Integral Equations 9 (1996),891-901.

[284] B. Hu and H.-M. Yin, Semilinear parabolic equations with prescribed energy,Rend. Circ. Mat. Palermo (2) 44 (1995), 479-505.

[285] J. Hulshof and R. van der Vorst, Differential systems with strongly indefinitevariational structure, J. Functional Analysis 114 (1993), 32-58.

[286] J. Huska, Periodic solutions in superlinear parabolic problems, Acta Math.Univ. Comenian. (N.S.) 71 (2002), 19-26.

[287] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equationsof parabolic and hyperbolic type, Hiroshima Math. J. 26 (1996), 475-491.

Bibliography 561

[288] K. Ishige and T. Kawakami, Asymptotic behavior of solutions for some semi-linear heat equations in R

N (2006), Preprint.

[289] K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equationwith large diffusion, J. Differential Equations 212 (2005), 114-128.

[290] H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinearequations, J. Differential Equations 26 (1977), 291-319.

[291] M.A. Jendoubi, A simple unified approach to some convergence theorems ofL. Simon, J. Funct. Anal. 153 (1998), 187-202.

[292] H. Jiang, Global existence of solutions on an activator-inhibitor model, Dis-crete Contin. Dyn. Syst. 14 (2006), 737-751.

[293] D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven bypositive sources, Arch. Rational Mech. Anal. 49 (1973), 241-269.

[294] I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems witha balance law and nonlinearities of exponential growth, J. Differential Equa-tions 165 (2000), 24-41.

[295] I. Kanel and M. Kirane, Global existence and large time behavior of positivesolutions to a reaction diffusion system, Differential Integral Equations 13(2000), 255-264.

[296] S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,Comm. Pure Appl. Math. 16 (1963), 305-330.

[297] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,Arch. Rational Mech. Anal. 58 (1975), 181-205.

[298] N. Kavallaris, A.A. Lacey and D. Tzanetis, Global existence and divergence ofcritical solutions of a non-local parabolic problem in Ohmic heating process,Nonlinear Anal. 58 (2004), 787-812.

[299] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusionequation, Ann. Inst. H. Poincare Analyse Non Lineaire 4 (1987), 423-452.

[300] O. Kavian, Introduction a la theorie des points critiques, Springer, Paris,1993.

[301] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equationwith subcritical nonlinearity, Ann. Inst. H. Poincare Anal. non lineaire 13(1996), 1-15.

[302] T. Kawanago, Existence and behavior of solutions for ut = ∆(um)+u , Adv.Math. Sci. Appl. 7 (1997), 367-400.

[303] H. Kawarada, On solutions of initial-boundary problem for ut = uxx +1/(1− u), Publ. Res. Inst. Math. Sci. 10 (1974/75), 729-736.

[304] B. Kawohl and L. Peletier, Observations on blow up and dead cores for non-linear parabolic equations, Math. Z. 202 (1989), 207-217.

562 Bibliography

[305] J.P. Keener and H.B. Keller, Positive solutions of convex nonlinear eigen-value problems, J. Differential Equations 16 (1974), 103-125.

[306] J.R. King, Personal communication 2006.[307] K. Kobayashi, T. Sirao and H. Tanaka, On the blowing up problem for semi-

linear heat equations, J. Math. Soc. Japan 29 (1977), 407-424.[308] A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, Etude de l’equation

de la diffusion avec croissance de la quantite de matiere et son applica-tion a un probleme biologique, Bulletin Universite d’Etat a Moscou (Bjul.Moskowskogo Gos. Univ.), Serie Internationale, Section A 1 (1937), 1-26.

[309] S. Kouachi, Existence of global solutions to reaction-diffusion systems via aLyapunov functional, Electron. J. Differential Equations 2001, 88 (2001),1-13.

[310] M.K. Kwong, Uniqueness of positive solutions of ∆u − u + up = 0 in Rn,

Arch. Rational Mech. Anal. 105 (1989), 243-266.[311] A.A. Lacey, Mathematical analysis of thermal runaway for spatially inho-

mogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350-1366.[312] A.A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc.

Roy. Soc. Edinburgh Sect. A 98 (1984), 183-202.[313] A.A. Lacey, Global blow-up of a nonlinear heat equation, Proc. Roy. Soc.

Edinburgh Sect. A 104 (1986), 161-167.[314] A.A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heat-

ing. I. Model derivation and some special cases, European J. Appl. Math. 6(1995), 127-144.

[315] A.A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heat-ing. II. General proof of blow-up and asymptotics of runaway, European J.Appl. Math. 6 (1995), 201-224.

[316] A.A. Lacey and D. Tzanetis, Global existence and convergence to a singularsteady state for a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect.A 105 (1987), 289-305.

[317] A.A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffusionequation with a sufficiently large initial condition, IMA J. Appl. Math. 41(1988), 207-215.

[318] A.A. Lacey and D. Tzanetis, Global, unbounded solutions to a parabolic equa-tion, J. Differential Equations 101 (1993), 80-102.

[319] O.A. Ladyzenskaja, Solution of the first boundary problem in the large forquasi-linear parabolic equations, Trudy Moskov. Mat. Obsc. 7 (1958), 149-177.

[320] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc., Transl. Math. Mono-graphs, Providence, R.I., 1968.

Bibliography 563

[321] Ph. Laurencot, Convergence to steady states for a one-dimensional vis-cous Hamilton-Jacobi equation with Dirichlet boundary conditions, PacificJ. Math. 230 (2007), 347-364.

[322] Ph. Laurencot and Ph. Souplet, On the growth of mass for a viscous Hamil-ton-Jacobi equation, J. Anal. Math. 89 (2003), 367-383.

[323] M. Lazzo, Solutions positives multiples pour une equation elliptique nonlineaire avec l’exposant critique de Sobolev, C. R. Acad. Sci. Paris Ser. IMath. 314 (1992), 61-64.

[324] T. Lee and W.-M. Ni, Global existence, large time behaviour and life spanof solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math.Soc. 333 (1992), 365-378.

[325] L.A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-con-duction equation with distributed parameters, Differentsial’nye Uravneniya24 (1988), 1226-1234, (English translation: Differential Equations 24 (1988),799-805).

[326] L.A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model.2 (1990), 63-74, (in Russian).

[327] H.A. Levine, Some nonexistence and instability theorems for solutions offormally parabolic equations of the form Put = −Au + F (u), Arch. RationalMech. Anal. 51 (1973), 371-386.

[328] H.A. Levine, The role of critical exponents in blowup theorems, SIAM Rev.32 (1990), 262-288.

[329] H.A. Levine, Quenching and beyond: a survey of recent results, Nonlinearmathematical problems in industry, II, GAKUTO Internat. Ser. Math. Sci.Appl., 2, Gakkotosho, Tokyo, 1993, pp. 501-512.

[330] H.A. Levine, P. Sacks, B. Straughan and L. Payne, Analysis of a convectivereaction-diffusion equation (II), SIAM J. Math. Anal. 20 (1989), 133-147.

[331] F. Li, S.-H. Huang and C.-H. Xie, Global existence and blow-up of solutionsto a nonlocal reaction-diffusion system, Discrete Contin. Dyn. Syst. 9 (2003),1519-1532.

[332] F. Li and M.-X. Wang, Properties of blow-up solutions to a parabolic systemwith nonlinear localized terms, Discrete Contin. Dyn. Syst. 13 (2005), 683-700.

[333] M. Li, S. Chen and Y. Qin, Boundedness and blow up for the general acti-vator-inhibitor model, Acta Math. Appl. Sinica (English Ser.) 11 (1995),59-68.

[334] Y. Li, Asymptotic behavior of positive solutions of equation ∆u+K(x)up = 0in R

n, J. Differential Equations 95 (1992), 304-330.[335] Y. Li and C.-H. Xie, Blow-up for semilinear parabolic equations with non-

linear memory, Z. Angew. Math. Phys. 55 (2004), 15-27.

564 Bibliography

[336] E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of twodomains, Invent. Math. 74 (1983), 441-448.

[337] G.M. Lieberman, The first initial-boundary value problem for quasilinearsecond order parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)13 (1986), 347-387.

[338] G.M. Lieberman, Second order parabolic differential equations, World Scien-tific, Singapore, 1996.

[339] Z. Lin, Blowup estimates for a mutualistic model in ecology, Electron. J.Qual. Theory Differential Equ. 2002, 8 (2002), 1-14.

[340] P.-L. Lions, Isolated singularities in semilinear problems, J. DifferentialEquations 38 (1980), 441-450.

[341] P.-L. Lions, Resolution de problemes elliptiques quasilineaires, Arch. Ratio-nal Mech. Anal. 74 (1980), 335-353.

[342] P.-L. Lions, Asymptotic behavior of some nonlinear heat equations, Phys. D5 (1982), 293-306.

[343] Y. Lou, Necessary and sufficient condition for the existence of positive solu-tions of certain cooperative system, Nonlinear Anal. 26 (1996), 1079-1095.

[344] A. Lunardi, Analytic semigroups and optimal regularity in parabolic prob-lems, Progress in Nonlinear Differential Equations and their Applications16, Birkhauser, Basel, 1995.

[345] Y. Martel, Complete blow up and global behaviour of solutions of ut−∆u =g(u), Ann. Inst. H. Poincare Anal. Non Lineaire 15 (1998), 687-723.

[346] Y. Martel and Ph. Souplet, Estimations optimales en temps petit et presde la frontiere pour les solutions de l’equation de la chaleur avec donneesnon-compatibles, C. R. Acad. Sci. Paris Ser. I Math. 79 (1998), 575-580.

[347] Y. Martel and Ph. Souplet, Small time boundary behavior for parabolic equa-tions with noncompatible data, J. Math. Pures Appl. 79 (2000), 603-632.

[348] R. Martin and M. Pierre, Nonlinear reaction-diffusion systems, Nonlinearequations in the applied sciences, Math. Sci. Engrg., 185, Academic Press,Boston, MA, 1992, pp. 363-398.

[349] R. Martin and M. Pierre, Influence of mixed boundary conditions in somereaction-diffusion systems, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997),1053-1066.

[350] K. Masuda, On the global existence and asymptotic behavior of solutions ofreaction-diffusion equations, Hokkaido Math. J. 12 (1983), 360-370.

[351] K. Masuda, Analytic solutions of some nonlinear diffusion equations, Math.Z. 187 (1984), 61-73.

[352] K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math. 4(1987), 47-58.

Bibliography 565

[353] H. Matano, Convergence of solutions of one-dimensional semilinear parabolicequations, J. Math. Kyoto Univ. 18 (1978), 221-227.

[354] H. Matano, Existence of nontrivial unstable sets for equilibriums of stronglyorder-preserving systems, J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 30 (1983),645-673.

[355] H. Matano and F. Merle, On nonexistence of type II blowup for a supercriticalnonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494-1541.

[356] H. Matano and F. Merle, in preparation.

[357] J. Matos, Blow up of critical and subcritical norms in semilinear heat equa-tions, Adv. Differential Equations 3 (1998), 497-532.

[358] J. Matos, Convergence of blow-up solutions of nonlinear heat equations in thesupercritical case, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 1197-1227.

[359] J. Matos, Unfocused blow up solutions of semilinear parabolic equations, Dis-crete Contin. Dyn. Syst. 5 (1999), 905-928.

[360] J. Matos, Self-similar blow up patterns in supercritical semilinear heat equa-tions, Comm. Appl. Anal. 5 (2001), 455-483.

[361] J. Matos and Ph. Souplet, Universal blow-up rates for a semilinear heatequation and applications, Adv. Differential Equations 8 (2003), 615-639.

[362] P.J. McKenna and W. Reichel, A priori bounds for semilinear equations anda new class of critical exponents for Lipschitz domains, J. Funct. Anal. 244(2007), 220-246.

[363] P. Meier, On the critical exponent for reaction-diffusion equations, Arch.Rational Mech. Anal. 109 (1990), 63-72.

[364] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-uppoints, Comm. Pure Appl. Math. 45 (1992), 263-300.

[365] F. Merle and L.A. Peletier, Positive solutions of elliptic equations involvingsupercritical growth, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 49-62.

[366] F. Merle and H. Zaag, Stability of the blow-up profile for equations of thetype ut = ∆u + |u|p−1u, Duke Math. J. 86 (1997), 143-195.

[367] F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior fornonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196.

[368] F. Merle and H. Zaag, Refined uniform estimates at blow-up and applicationsfor nonlinear heat equations, Geom. Funct. Anal. 8 (1998), 1043-1085.

[369] F. Mignot and J.-P. Puel, Sur une classe de problemes non lineaires avecune non-linearite positive, croissante, convexe, Comm. Partial DifferentialEquations 5 (1980), 791-836.

[370] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differ-ential Equations 18 (1993), 125-151.

566 Bibliography

[371] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systemsin R

n, Differential Integral Equations 9 (1996), 465-479.

[372] E. Mitidieri and S.I. Pohozaev, A priori estimates and blow-up of solutions ofnonlinear partial differential equations and inequalities, Proc. Steklov Inst.Math. 234 (2001), 1-362.

[373] E. Mitidieri and S.I. Pohozaev, Fujita-type theorems for quasilinear parabolicinequalities with a nonlinear gradient, Dokl. Akad. Nauk 386 (2002), 160-164.

[374] N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equationwith supercritical nonlinearity, Math. Z. 239 (2002), 215-229.

[375] N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation withthe Neumann boundary condition, J. Differential Equations 193 (2003), 212-238.

[376] N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equationwith supercritical nonlinearity, J. Differential Equations 205 (2004), 298-328.

[377] N. Mizoguchi, Type-II blowup for a semilinear heat equation, Adv. Differen-tial Equations 9 (2004), 1279-1316.

[378] N. Mizoguchi, Boundedness of global solutions for a supercritical semilinearheat equation and its application, Indiana Univ. Math. J. 54 (2005), 1047-1059.

[379] N. Mizoguchi, Various behaviors of solutions for a semilinear heat equationafter blowup, J. Funct. Anal. 220 (2005), 214-227.

[380] N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation,Math. Ann. 331 (2005), 461-473.

[381] N. Mizoguchi, Growup of solutions for a semilinear heat equation with su-percritical nonlinearity, J. Differential Equations 227 (2006), 652-669.

[382] N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation II,J. Differential Equations 231 (2006), 182-194.

[383] N. Mizoguchi, H. Ninomiya and E. Yanagida, Diffusion induced blow-up ina nonlinear parabolic system, J. Dynam. Differential Equations 10 (1998),619-638.

[384] N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutionswith sign changes in a semilinear parabolic equation, Math. Ann. 307 (1997),663-675.

[385] N. Mizoguchi and E. Yanagida, Critical exponents for the blowup of solu-tions with sign changes in a semilinear parabolic equation. II, J. DifferentialEquations 145 (1998), 295-331.

[386] N. Mizoguchi and E. Yanagida, Blow-up and life span of solutions for asemilinear parabolic equation, SIAM J. Math. Anal. 29 (1998), 1434-1446.

Bibliography 567

[387] J. Morgan, On a question of blow-up for semilinear parabolic systems, Dif-ferential Integral Equations 3 (1990), 973-978.

[388] C.E. Mueller and F.B. Weissler, Single point blow-up for general semilinearheat equation, Indiana Univ. Math. J. 34 (1985), 881-913.

[389] Y. Naito, Non-uniqueness of solutions to the Cauchy problem for semilinearheat equations with singular initial data, Math. Ann. 329 (2004), 161-196.

[390] Y. Naito, An ODE approach to the multiplicity of self-similar solutions forsemi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006),807-835.

[391] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, No-tices Amer. Math. Soc. 45 (1998), 9-18.

[392] W.-M. Ni, Qualitative properties of solutions to elliptic problems, Handbookof Differential Equations, Stationary partial differential equations. Vol. I (M.Chipot et al., eds.), Elsevier/North-Holland, Amsterdam, 2004, pp. 157-233.

[393] W.-M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positiveradial solutions of ∆u + f(u, r) = 0, Comm. Pure Appl. Math. 38 (1985),67-108.

[394] W.-M. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations,Trans. Amer. Math. Soc. 287 (1985), 657-671.

[395] W.-M. Ni and P. Sacks, The number of peaks of positive solutions of semi-linear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460-471.

[396] W.-M. Ni, P. Sacks and J. Tavantzis, On the asymptotic behavior of solutionsof certain quasilinear parabolic equations, J. Differential Equations 54 (1984),97-120.

[397] W.-M. Ni and J. Serrin, Existence and nonexistence theorems for groundstates of quasilinear partial differential equations. The anomalous case, AttiAccad. Naz. Lincei 77 (1986), 231-257.

[398] W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations 229 (2006), 426-465.

[399] S.M. Nikolskii, Approximation of functions of several variables and imbeddingtheorems, Springer, Berlin - Heidelberg - New York, 1975.

[400] R.D. Nussbaum, Positive solutions of nonlinear elliptic boundary value prob-lems, J. Math. Anal. Appl. 51 (1975), 461-482.

[401] A. Okada and I. Fukuda, Total versus single point blow-up of solutions ofa semilinear parabolic equation with localized reaction, J. Math. Anal. Appl.281 (2003), 485-500.

[402] O. Oleinik and S. Kruzkov, Quasi-linear parabolic second order equationswith several independent variables, Uspehi Mat. Nauk 16 (1961), 115-155.

568 Bibliography

[403] M. Otani, Existence and asymptotic stability of strong solutions of nonlinearevolution equations with a difference term of subdifferentials, Colloq. Math.Soc. Janos Bolyai 30, North-Holland, Amsterdam-New York, 1981, pp. 795-809.

[404] F. Pacard, Existence and convergence of positive weak solutions of −∆u =u

nn−2 in bounded domains of R

n, n ≥ 3, Calc. Var. Partial Differential Equa-tions 1 (1993), 243-265.

[405] C.-V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, NewYork, 1992.

[406] D. Passaseo, Nonexistence results for elliptic problems with supercritical non-linearity in nontrivial domains, J. Funct. Anal. 114 (1993), 97-105.

[407] D. Passaseo, New nonexistence results for elliptic equations with supercriticalnonlinearity, Differential Integral Equations 8 (1995), 577-586.

[408] D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equationswith critical Sobolev exponent in some contractible domains, ManuscriptaMath. 65 (1989), 147-165.

[409] L.E. Payne and D.H. Sattinger, Saddle points and instability of nonlinearhyperbolic equations, Israel J. Math. 22 (1975), 273-303.

[410] A. Pazy, Semigroups of linear operators and applications to partial differen-tial equations, Springer, New York, 1983.

[411] M. Pedersen and Z. Lin, Coupled diffusion systems with localized nonlinearreactions, Comput. Math. Appl. 42 (2001), 807-816.

[412] L.A. Peletier, D. Terman and F.B. Weissler, On the equation ∆u + x · ∇u +f(u) = 0, Arch. Rational Mech. Anal. 94 (1986), 83-99.

[413] L.A. Peletier and R. van der Vorst, Existence and nonexistence of positivesolutions of nonlinear elliptic systems and the biharmonic equation, Differ-ential Integral Equations 5 (1992), 747-767.

[414] M. Pierre (2003), Personal communication.

[415] M. Pierre, Weak solutions and supersolutions in L1 for reaction-diffusionsystems, J. Evol. Equ. 3 (2003), 153-168.

[416] M. Pierre and D. Schmitt, Global existence for a reaction-diffusion systemwith a balance law, Semigroups of linear and nonlinear operations and ap-plications Math. Sci. Engrg., 185, Kluwer Acad. Publ., Dordrecht, 1993,pp. 251-258.

[417] M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissi-pation of mass, SIAM J. Math. Anal. 28 (1997), 259-269.

[418] P. Plechac and V. Sverak, Singular and regular solutions of a nonlinearparabolic system, Nonlinearity 16 (2003), 2083-2097.

Bibliography 569

[419] S.I. Pohozaev, Eigenfunctions of the equation ∆u+λf(u) = 0, Soviet Math.Dokl. 5 (1965), 1408-1411.

[420] P. Polacik, Morse indices and bifurcations of positive solutions of ∆u+f(u) =0 on R

N , Indiana Univ. Math. J. 50 (2001), 1407-1432.

[421] P. Polacik, Parabolic equations: Asymptotic behavior and dynamics on in-variant manifolds, Handbook of dynamical systems, Vol. 2 (B. Fiedler, ed.),Elsevier, Amsterdam, 2002, pp. 835-883.

[422] P. Polacik and P. Quittner, A Liouville-type theorem and the decay of radialsolutions of a semilinear heat equation, Nonlinear Anal. 64 (2006), 1679-1689.

[423] P. Polacik and P. Quittner, in preparation.

[424] P. Polacik, P. Quittner and Ph. Souplet, Singularity and decay estimates insuperlinear problems via Liouville-type theorems. Part I: elliptic equationsand systems, Duke Math. J. 139 (2007) (to appear).

[425] P. Polacik, P. Quittner and Ph. Souplet, Singularity and decay estimates insuperlinear problems via Liouville-type theorems. Part II: parabolic equations,Indiana Univ. Math. J. 56 (2007), 879-908.

[426] P. Polacik and K.P. Rybakowski, Nonconvergent bounded trajectories insemilinear heat equations, J. Differential Equations 124 (1996), 472-494.

[427] P. Polacik and F. Simondon, Nonconvergent bounded solutions of semilinearheat equations on arbitrary domains, J. Differential Equations 186 (2002),586-610.

[428] P. Polacik and E. Yanagida, On bounded and unbounded global solutions ofa supercritical semilinear heat equation, Math. Ann. 327 (2003), 745-771.

[429] M. Protter and H. Weinberger, Maximum principles in differential equations,Prentice Hall, Englewood Cliffs, N.J., 1967.

[430] J. Pruss and H. Sohr, Imaginary powers of elliptic second order differentialoperators in Lp-spaces, Hiroshima Math. J. 23 (1993), 161-192.

[431] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math.J. 35 (1986), 681-703.

[432] F. Quiros and J. Rossi, Non-simultaneous blow-up in a semilinear parabolicsystem, Z. Angew. Math. Phys. 52 (2001), 342-346.

[433] P. Quittner, Blow-up for semilinear parabolic equations with a gradient term,Math. Methods Appl. Sci. 14 (1991), 413-417.

[434] P. Quittner, On global existence and stationary solutions for two classes ofsemilinear parabolic equations, Comment. Math. Univ. Carolin. 34 (1993),105-124.

[435] P. Quittner, Global existence of solutions of parabolic problems with nonlinearboundary conditions, Banach Center Publ. 33 (1996), 309-314.

570 Bibliography

[436] P. Quittner, Signed solutions for a semilinear elliptic problem, DifferentialIntegral Equations 11 (1998), 551-559.

[437] P. Quittner, A priori bounds for global solutions of a semilinear parabolicproblem, Acta Math. Univ. Comenian. (N.S.) 68 (1999), 195-203.

[438] P. Quittner, Universal bound for global positive solutions of a superlinearparabolic problem, Math. Ann. 320 (2001), 299-305.

[439] P. Quittner, A priori estimates of global solutions and multiple equilibria ofa parabolic problem involving measure, Electron. J. Differential Equations2001, 29 (2001), 1-17.

[440] P. Quittner, Continuity of the blow-up time and a priori bounds for solutionsin superlinear parabolic problems, Houston J. Math. 29 (2003), 757-799.

[441] P. Quittner, Multiple equilibria, periodic solutions and a priori bounds forsolutions in superlinear parabolic problems, NoDEA Nonlinear DifferentialEquations Appl. 11 (2004), 237-258.

[442] P. Quittner, Complete and energy blow-up in superlinear parabolic problems,Recent Advances in Elliptic and Parabolic Problems (Chiun-Chuan Chen,Michel Chipot and Chang-Shou Lin, eds.), World Scientific Publ., Hacken-sack, NJ, 2005, pp. 217-229.

[443] P. Quittner, The decay of global solutions of a semilinear parabolic equation,Discrete Contin. Dyn. Syst. (to appear).

[444] P. Quittner, Qualitative theory of semilinear parabolic equations and sys-tems, Lectures on evolutionary partial differential equations, Lecture notesof the Jindrich Necas Center for Mathematical Modeling, vol. 2, Matfyzpress,Praha, 2007 (to appear).

[445] P. Quittner and A. Rodrıguez-Bernal, Complete and energy blow-up in par-abolic problems with nonlinear boundary conditions, Nonlinear Anal. 62(2005), 863-875.

[446] P. Quittner and Ph. Souplet, Admissible Lp norms for local existence andfor continuation in semilinear parabolic systems are not the same, Proc. Roy.Soc. Edinburgh Sect. A 131 (2001), 1435-1456.

[447] P. Quittner and Ph. Souplet, Global existence from single-component Lp

estimates in a semilinear reaction-diffusion system, Proc. Amer. Math. Soc.130 (2002), 2719-2724.

[448] P. Quittner and Ph. Souplet, A priori estimates of global solutions of su-perlinear parabolic problems without variational structure, Discrete Contin.Dyn. Syst. 9 (2003), 1277-1292.

[449] P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic sys-tems via bootstrap in weighted Lebesgue spaces, Arch. Rational Mech. Anal.174 (2004), 49-81.

Bibliography 571

[450] P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universalbounds for nonlinear heat equations, J. Differential Equations 196 (2004),316-339.

[451] P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, RockyMountain J. Math. 3 (1973), 161-202.

[452] P.H. Rabinowitz, Multiple critical points of perturbed symmetric functionals,Trans. Amer. Math. Soc. 272 (1982), 753-770.

[453] M. Ramos, S. Terracini and C. Troestler, Superlinear indefinite elliptic prob-lems and Pohozaev type identities, J. Funct. Anal. 159 (1998), 596-628.

[454] W. Reichel and H. Zou, Non-existence results for semilinear cooperative el-liptic systems via moving spheres, J. Differential Equations 161 (2000), 219-243.

[455] F. Rellich, Darstellung der Eigenwerte von ∆u + λu = 0 durch ein Randin-tegral, Math. Z. 46 (1940), 635-636.

[456] O. Rey, A multiplicity result for a variational problem with lack of compact-ness, Nonlinear Anal. 13 (1989), 1241-1249.

[457] F. Ribaud, Analyse de Littlewood Paley pour la resolution d’equations para-boliques semi-lineaires, Doctoral Thesis, University of Paris XI, 1996.

[458] J. Rossi and Ph. Souplet, Coexistence of simultaneous and nonsimultaneousblow-up in a semilinear parabolic system, Differential Integral Equations 18(2005), 405-418.

[459] F. Rothe, Uniform bounds from bounded Lp-functionals in reaction-diffusionequations, J. Differential Equations 45 (1982), 207-233.

[460] F. Rothe, Global solutions of reaction-diffusion systems, Lecture Notes inMathematics 1072, Springer-Verlag, Berlin - Heidelberg - New York, 1984.

[461] P. Rouchon, Blow-up of solutions of nonlinear heat equations in unboundeddomains for slowly decaying initial data, Z. Angew. Math. Phys. 52 (2001),1017-1032.

[462] P. Rouchon, Boundedness of global solutions for nonlinear diffusion equationswith localized source, Differential Integral Equations 16 (2003), 1083-1092.

[463] P. Rouchon, Universal bounds for global solutions of a diffusion equationwith a nonlocal reaction term, J. Differential Equations 193 (2003), 75-94.

[464] P. Rouchon, Universal bounds for global solutions of a diffusion equation witha mixed local-nonlocal reaction term, Acta Math. Univ. Comenian. (N.S.) 75(2006), 63-74.

[465] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations andnucleation, IMA J. Appl. Math. 48 (1992), 249-264.

[466] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov,Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987, Englishtranslation: Walter de Gruyter, Berlin, 1995.

572 Bibliography

[467] D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch.Rational Mech. Math. 30 (1968), 148-172.

[468] D. Schmitt, Existence globale ou explosion pour les systemes de reaction-diffusion avec controle de masse, Doctoral Thesis, University of Nancy 1,1995.

[469] J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolicequations, Contributions to nonlinear functional analysis, Academic Press,New York, 1971, pp. 565-601.

[470] J. Serrin and H. Zou, Existence and non-existence for ground states of quasi-linear elliptic equations, Arch. Rational Mech. Anal. 121 (1992), 101-130.

[471] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emdensystems, Differential Integral Equations 9 (1996), 635-653.

[472] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emdensystem, Atti Sem. Mat. Fis. Univ. Modena 46 (1998, suppl.), 369-380.

[473] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theoremsfor quasilinear elliptic equations and inequalities, Acta Math. 189 (2002),79-142.

[474] L. Simon, Asymptotics for a class of non-linear evolution equations, withapplications to geometric problems, Ann. Math. 118 (1983), 525-571.

[475] S. Snoussi, S. Tayachi and F.B. Weissler, Asymptotically self-similar globalsolutions of a general semilinear heat equation, Math. Ann. 321 (2001), 131-155.

[476] S. Sohr, Beschrankter H∞-Funktionalkalkul fur elliptische Randwertsysteme,Dissertation, Kassel 1999.

[477] V. Solonnikov, Green matrices for parabolic boundary value problems, Zap.Naucn. Sem. Leningrad 14 (1969), 132-150.

[478] Ph. Souplet, Sur l’asymptotique des solutions globales pour une equation dela chaleur semi-lineaire dans des domaines non bornes, C. R. Acad. Sci.Paris Ser. I Math. 323 (1996), 877-882.

[479] Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J.Math. Anal. 28 (1998), 1301-1334.

[480] Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffu-sion equations with nonlocal nonlinear source, J. Differential Equations 153(1999), 374-406.

[481] Ph. Souplet, Geometry of unbounded domains, Poincare inequalities and sta-bility in semilinear parabolic equations, Comm. Partial Differential Equations24 (1999), 951-973.

[482] Ph. Souplet, A note on the paper by Qi S. Zhang: “A priori estimates andthe representation formula for all positive solutions to a semilinear parabolicproblem”, J. Math. Anal. Appl. 243 (2000), 453-457.

Bibliography 573

[483] Ph. Souplet, Decay of heat semigroups in L∞ and applications to nonlinearparabolic problems in unbounded domains, J. Funct. Anal. 173 (2000), 343-360.

[484] Ph. Souplet, Recent results and open problems on parabolic equations withgradient nonlinearities, Electron. J. Differential Equations 2001, 20 (2001),1-19.

[485] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic e-quations with general boundary conditions, Differential Integral Equations15 (2002), 237-256.

[486] Ph. Souplet, Monotonicity of solutions and blow-up for semilinear parabolicequations with nonlinear memory, Z. Angew. Math. Phys. 55 (2004), 28-31.

[487] Ph. Souplet, Uniform blow-up profile and boundary behaviour for a non-localreaction-diffusion equation with critical damping, Math. Methods Appl. Sci.27 (2004), 1819-1829.

[488] Ph. Souplet, Infinite time blow-up for superlinear parabolic problems withlocalized reaction, Proc. Amer. Math. Soc. 133 (2005), 431-436.

[489] Ph. Souplet, Optimal regularity conditions for elliptic problems via Lpδ spaces,

Duke Math. J. 127 (2005), 175-192.

[490] Ph. Souplet, The influence of gradient perturbations on blow-up asymptoticsin semilinear parabolic problems: a survey, Progress in Nonlinear DifferentialEquations and Their Applications, Vol. 64, Birkhauser, 2005, pp. 473-496.

[491] Ph. Souplet, A remark on the large-time behavior of solutions of viscousHamilton-Jacobi equations, Acta Math. Univ. Comenian. (N.S.) 76 (2007),11-13.

[492] Ph. Souplet, A note on diffusion-induced blow-up, J. Dynam. DifferentialEquations 19 (2007), to appear.

[493] Ph. Souplet, Single point blow-up for a semilinear parabolic system, J. Eur.Math. Soc. (to appear).

[494] Ph. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-upin a reaction-diffusion system, J. Math. Soc. Japan 56 (2004), 571-584.

[495] Ph. Souplet, S. Tayachi and F.B. Weissler, Exact self-similar blow-up ofsolutions of a semilinear parabolic equation with a nonlinear gradient term,Indiana Univ. Math. J. 45 (1996), 655-682.

[496] Ph. Souplet and J.L. Vazquez, Stabilization towards a singular steady statewith gradient blow-up for a convection-diffusion problem, Discrete Contin.Dyn. Syst. 14 (2006), 221-234.

[497] Ph. Souplet and F.B. Weissler, Self-similar subsolutions and blowup for non-linear parabolic equations, J. Math. Anal. Appl. 212 (1997), 60-74.

574 Bibliography

[498] Ph. Souplet and F.B. Weissler, Poincare’s inequality and global solutions ofa nonlinear parabolic equation, Ann. Inst. H. Poincare Anal. Non Lineaire16 (1999), 337-373.

[499] Ph. Souplet and F.B. Weissler, Regular self-similar solutions of the nonlinearheat equation with initial data above the singular steady state, Ann. Inst. H.Poincare Anal. Non Lineaire 20 (2003), 213-235.

[500] Ph. Souplet and Q.S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math. 99 (2006), 355-396.

[501] M.A.S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differential Integral Equations 8 (1995),1245-1258.

[502] R. Sperb, Growth estimates in diffusion-reaction problems, Arch. RationalMech. Anal. 75 (1980), 127-145.

[503] P.N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Dif-ferential Integral Equations 6 (1993), 663-670.

[504] B. Straughan, Explosive instabilities in mechanics, Springer, Berlin, 1998.

[505] M. Struwe, Variational methods. Applications to nonlinear partial differentialequations and Hamiltonian systems, Springer, Berlin, 2000.

[506] M. Struwe, Infinitely many critical points for functionals which are not evenand applications to nonlinear boundary value problems, Manuscripta Math.32 (1980), 335-364.

[507] R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equationswith slowly decaying initial data, Adv. Math. Sci. Appl. 9 (1999), 291-317.

[508] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. (4)110 (1976), 353-372.

[509] S. Taliaferro, Local behavior and global existence of positive solutions ofauλ ≤ −∆u ≤ uλ, Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002),889-901.

[510] J.I. Tello, Stability of steady states of the Cauchy problem for the exponentialreaction-diffusion equation, J. Math. Anal. Appl. 324 (2006), 381-396.

[511] E. Terraneo, Non-uniqueness for a critical non-linear heat equation, Comm.Partial Differential Equations 27 (2002), 185-218.

[512] Al. Tersenov and Ar. Tersenov, Global solvability for a class of quasilinearparabolic problems, Indiana Univ. Math. J. 50 (2001), 1899-1913.

[513] H. Triebel, Interpolation theory, function spaces, differential operators,North-Holland, Amsterdam - New York - Oxford, 1978.

[514] W.C. Troy, Symmetry properties in systems of semilinear elliptic equations,J. Differential Equations 42 (1981), 400-413.

Bibliography 575

[515] M. Tsutsumi, On solutions of semilinear differential equations in a Hilbertspace, Math. Japon. 17 (1972), 173-193.

[516] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc.London Ser. B 237 (1952), 37-72.

[517] R.E.L. Turner, A priori bounds for positive solutions of nonlinear ellipticequations in two variables, Duke Math. J. 41 (1974), 759-774.

[518] D. Tzanetis, Blow-up of radially symmetric solutions of a non-local problemmodelling Ohmic heating, Electron. J. Differential Equations (2002, 11), 1-26.

[519] J.J.L. Velazquez, Local behaviour near blow-up points for semilinear parabolicequations, J. Differential Equations 106 (1993), 384-415.

[520] J.J.L. Velazquez, Estimates on the (n − 1)-dimensional Hausdorff measureof the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42(1993), 445-476.

[521] J.J.L. Velazquez, Blow up for semilinear parabolic equations, Recent ad-vances in partial differential equations (Res. Appl. Math. 30) (M.A. Herrero,E. Zuazua, eds.), Masson, Paris, 1994, pp. 131-145.

[522] J.J.L. Velazquez, V.A. Galaktionov and M.A. Herrero, The space structurenear a blow-up point for semilinear heat equations: a formal approach, Zh.Vychisl. Mat. i Mat. Fiziki 31 (1991), 399-411.

[523] L. Veron, Singularities of solutions of second order quasilinear equations,Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996.

[524] L. Wang and Q. Chen, The asymptotic behaviour of blow-up solution oflocalized nonlinear equation, J. Math. Anal. Appl. 200 (1996), 315-321.

[525] M.-X. Wang and Y. Wang, Properties of positive solutions for non-localreaction-diffusion problems, Math. Methods Appl. Sci. 19 (1996), 1141-1156.

[526] N.A. Watson, Parabolic equations on an infinite strip, Marcel Dekker, Bos-ton, 1989.

[527] H. Weinberger, An example of blowup produced by equal diffusions, J. Dif-ferential Equations 154 (1999), 225-237.

[528] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct.Anal. 32 (1979), 277-296.

[529] F.B. Weissler, Local existence and nonexistence for semilinear parabolic equa-tions in Lp, Indiana Univ. Math. J. 29 (1980), 79-102.

[530] F.B. Weissler, Existence and non-existence of global solutions for a semilin-ear heat equation, Israel J. Math. 38 (1981), 29-40.

[531] F.B. Weissler, Single point blow-up for a semilinear initial value problem, J.Differential Equations 55 (1984), 204-224.

576 Bibliography

[532] F.B. Weissler, Asymptotic analysis of an ordinary differential equation andnon-uniqueness for a semilinear partial differential equation, Arch. RationalMech. Anal. 91 (1985), 231-245.

[533] F.B. Weissler, Rapidly decaying solutions of an ordinary differential equa-tion with applications to semilinear elliptic and parabolic partial differentialequations, Arch. Rational Mech. Anal. 91 (1985), 247-266.

[534] F.B. Weissler, An L∞ blow-up estimate for a nonlinear heat equation, Comm.Pure Appl. Math. 38 (1985), 291-295.

[535] F.B. Weissler, Lp-energy and blow-up for a semilinear heat equation, In Non-linear functional analysis and its applications, Proc. Symp. Pure Math. 45/2(1986), 545-551.

[536] B. Wollenmann, Uniqueness for semilinear parabolic problems, PhD Thesis,Universitat Zurich, 2001.

[537] R. Xing, A priori estimates for global solutions of semilinear heat equationsin R

n, Nonlinear Anal. (to appear).[538] E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weiss-

ler equation, J. Differential Equations 127 (1996), 561-570.[539] X.-F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic

PDEs, J. Funct. Anal. 160 (1998), 223-253.[540] H. Zaag, A Liouville theorem and blow-up behavior for a vector-valued non-

linear heat equation with no gradient structure, Comm. Pure Appl. Math.54 (2001), 107-133.

[541] H. Zaag, On the regularity of the blow-up set for semilinear heat equations,Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002), 505-542.

[542] H. Zaag, Determination of the curvature of the blow-up set and refined sin-gular behavior for a semilinear heat equation, Duke Math. J. 133 (2006),499-525.

[543] T.I. Zelenyak, Stabilization of solutions of boundary value problems for a sec-ond order parabolic equation with one space variable, Differential Equations4 (1968), 17-22.

[544] L. Zhang, Uniqueness of positive solutions of ∆u + u + up = 0 in a ball,Comm. Partial Differential Equations 17 (1992), 1141-1164.

[545] Q.S. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians,J. Differential Equations 182 (2002), 416-430.

[546] P. Zhao and C. Zhong, On the infinitely many positive solutions of a super-critical elliptic problem, Nonlinear Anal. 44 (2001), 123-139.

[547] H. Zou, A priori estimates for a semilinear elliptic systems without varia-tional structure and their applications, Math. Ann. 323 (2002), 713-735.

List of Symbols

Standard function spaces are defined in Preliminaries.

BR, BR(x), B(x, R) . . . . . . . . . . . 1

Sn−1 . . . . . . . . . . . . . . . . . . . . . . . . . 1

χM . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

D′ ⊂⊂ D . . . . . . . . . . . . . . . . . . . . . 1

s+, s− . . . . . . . . . . . . . . . . . . . . . . . . 1

R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

δ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 1

ν(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 1

QT , ST , PT . . . . . . . . . . . . . . . . . . .1

X ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

p′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

X → Y . . . . . . . . . . . . . . . . . . . . . . .2

X →→ Y . . . . . . . . . . . . . . . . . . . . 2

X.= Y . . . . . . . . . . . . . . . . . . . . . . . .2

L(X, Y ) . . . . . . . . . . . . . . . . . . . . . . 2

‖ · ‖k,p . . . . . . . . . . . . . . . . . . . . . . . . 2

‖ · ‖p . . . . . . . . . . . . . . . . . . . . . . . . . 2

‖ · ‖p,δ . . . . . . . . . . . . . . . . . . . . . . . . 3

Lpul . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

‖ · ‖p,ul . . . . . . . . . . . . . . . . . . . . . . . 3

·α . . . . . . . . . . . . . . . . . . . . . . . . . . 4

‖ · ‖2,1;p . . . . . . . . . . . . . . . . . . . . . . .4

[·]α;Q . . . . . . . . . . . . . . . . . . . . . . . . . 4

| · |a;Q . . . . . . . . . . . . . . . . . . . . . . . . 4

λk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

ϕk . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

e−tA . . . . . . . . . . . . . . . . . . . . . . . . . .5

δy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

pS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

psg . . . . . . . . . . . . . . . . . . . . . . . . . . 11

U∗ . . . . . . . . . . . . . . . . . . . . . . . . . . .11

Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . . 36

pJL . . . . . . . . . . . . . . . . . . . . . . . . . .50

pBT . . . . . . . . . . . . . . . . . . . . . . . . . 56

qc . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

pF . . . . . . . . . . . . . . . . . . . . . . . . . .101

pB . . . . . . . . . . . . . . . . . . . . . . . . . 137

pL . . . . . . . . . . . . . . . . . . . . . . . . . .167

B(u0) . . . . . . . . . . . . . . . . . . . . . . 190

X , X+ (Chapter III) . . . . . . . .272

X , X+ (Chapter IV) . . . . . . . .314

Lqg . . . . . . . . . . . . . . . . . . . . . . . . . .434

Hkg . . . . . . . . . . . . . . . . . . . . . . . . . 434

GΩ(x, y, t), G(x, y, t) . . . . . . . .440

G(x, t), Gt(x) . . . . . . . . . . . . . . .440

KΩ(x, y), K(x, y) . . . . . . . . . . . 440

‖ · ‖A . . . . . . . . . . . . . . . . . . . . . . .466

ω(−A), σ(−A) . . . . . . . . . . . . . .467

Xθ . . . . . . . . . . . . . . . . . . . . . . . . . 467

(·, ·)θ, [·, ·]θ, (·, ·)θ,p . . . . . . . . . 467

| · |α . . . . . . . . . . . . . . . . . . . . . . . . 468

z[0,R](ψ) . . . . . . . . . . . . . . . . . . . .526

Index

Lp-Lq-estimates– elliptic, 431– parabolic, 440Lp-estimates– elliptic, 429– parabolic, 438Lp

δ-Lqδ-estimates, 451

Lpδ-spaces, 2, 61, 86, 203, 266, 447

ω-limit set, 122, 123, 138, 197, 199,234, 237, 244, 312, 528

a priori estimates, a priori bounds– applications of, 230– elliptic, 55, 61, 65, 68, 251– of global solutions (parabolic), 161,

166, 202, 280, 338, 359, 395, 412– parabolic, 83, 150absorption, 203analytic semigroup, 466annulus, 21, 25, 26auxiliary function J

– for blow-up rate, 180, 286, 340, 407,424

– for space profile, 190, 342, 344, 348,400, 409

barrier, 316, 323, 388, 452Bernstein technique, 316, 363, 367,

376bifurcation diagram, 26, 28, 55blow-up– above a positive equilibrium, 96, 151– at infinity, 193– complete, 176, 219

– diffusion-induced, 301– for slow decay initial data, 98, 321– global, 194, 219, 381, 398, 413– incomplete, 227, 229– nonsimultaneous, 287– regional, 194, 354– single-point, 190, 193, 194, 286, 302,

348, 354, 375, 398, 403, 427blow-up criterion, 91, 94, 96, 98, 319,

321blow-up profile, 190, 201, 274, 286,

348, 364, 398, 403, 427– uniform, 381, 398blow-up rate, 177, 195, 283, 286, 338,

367, 381, 398, 404, 418, 424– applications, 189– initial, 202– refined estimate, 180– type I/type II, 179, 184, 340– universal estimate, 209, 211, 338blow-up set, 190, 286, 302, 329, 348,

375, 381, 398, 403, 413, 427– regularity of, 194blow-up time, see existence timebootstrap– alternate (elliptic systems), 266– elliptic, 9, 11, 14, 17, 39, 61, 72, 258,

293, 431– parabolic, 81, 83, 163, 195, 235, 348,

352, 475, 493, 502, 506boundary layer, 373, 381, 418bounded imaginary powers, 468, 491

cap, 45Caratheodory function, 7

580 Index

comparison principle, 315, 507, 509– nonlocal problems, 377, 404, 414,

419, 523– systems, 272, 288, 523concavity method, 94continuation after blow-up, 227, 373continuation property, 89, 273continuous dependence, 471, 473, 476,

478, 498convection term, 313convex– domain, 25, 68, 179, 180, 193, 195,

211, 354– nonlinearity, 92, 120, 121, 125, 221critical exponents, 314, 348, 360– for blow-up, 319, 321, 323, 330, 403,

404, 422– for scaling properties, 330– for well-posedness, 76, 86, 113, 190,

230, 394– Fujita, 101, 334, 418– Sobolev, 7critical value, 29, 31

dead-core, 341decay– of initial data, 98, 131, 132, 137, 239– time rate, 113, 132, 136, 142, 210,

239, 240, 246, 323, 358, 378, 441,485, 489

deformation lemma, 30degree– Brouwer, 69– Leray-Schauder, 58diffusion– eliminating blow-up, 125, 311– large/small diffusion limit, 194– preserving global existence, 288, 298Dirac delta distribution, 13, 79

dissipative gradient term, 313distance– parabolic, 213– to the boundary (see also Lp

δ-spa-ces), 1, 452

domain, 1domain of attraction, 120, 174, 532doubling lemma, 40, 211, 339, 401duality argument, 290, 292, 293, 515dynamical systems (see also ω-limit

set), 197, 528

eigenfunction method, 91, 174, 216,319, 321, 357, 422

eigenfunction, eigenvalue, 5, 26, 52,99, 104, 137, 194, 437, 486

– first, 5eigenvalue problem (nonlinear), 12, 28energy, 29, 51, 89, 163, 165, 192, 206,

234, 347, 499– functional, 9, 20, 93, 162, 395, 411– space, 137– weighted, 181, 199existence time, 87, 471, 497, 498– continuity, 176, 189, 224, 228, 360– estimate, 92, 178, 179, 407expanding wave, 326exponential nonlinearity, 171extremal point, 121

Fisher-KPP equation, 109fixed-point theorem– Banach, 50, 79, 471, 480, 495– Schauder, 398Fujita-type theorem, 100, 215, 280,

330, 418fundamental solution, 14, 432

Index 581

GBU, see gradient blow-upglobal existence– below the singular steady-state, 131– for small data, 112, 129, 141global solutions, 89, 528– asymptotic profiles, 139, 362– boundedness of, 161, 166, 169, 171,

188, 359, 360, 393, 412– decay of, 137, 202, 210– structure of, 120– unbounded, 171, 184, 246, 328, 340,

360, 404, 414, 422– weak, 227, 304gradient blow-up (GBU), 100, 314,

355, 374gradient estimate, gradient bound– elliptic, 43– parabolic, 153, 189, 314, 323, 356,

363, 442, 452gradient nonlinearities, 313grow-up rates, 174, 246

Holder continuous, 3– locally, 3– locally α-, 3hair-trigger effect, 109half-space, 36, 37, 67, 108, 156, 209,

260, 261, 265homogeneous– initial data, 141, 147– nonlinearities, 20Hopf’s lemma– elliptic, 454, 507– parabolic, 511Hopf-Cole transformation, 317, 361

identity– Bochner, 40

– Pohozaev, 18, 22, 25, 26, 35, 68, 123,174, 198, 235, 238, 253

– Rellich-Pohozaev type, 20indefinite coefficients, 67, 230inequality– Gagliardo-Nirenberg, 115– Holder, 462– Hardy, 465– Hardy-Sobolev, 55, 271, 466– Harnack, 216– interpolation, 200, 362, 462– Jensen, 462– Poincare, 113, 323, 325, 463– singular Gronwall, 470, 505– Sobolev, 462– Sobolev, best constant, 23– Young, 44, 462– Young (for convolutions), 433initial blow-up rate, 202initial trace, 79inradius, 113, 323, 463instability– of equilibria, 113, 122– of the blow-up rate, 340instantaneous attractors, 203interpolation, 163– couple, 469– embedding, 5– functor, 467, 468, 488– inequality, 200, 362, 462interpolation-extrapolation spaces, 76,

90, 466intersection-comparison, see zero num-

berinvariance principle, 530invariant space, 141isolated singularity, 12, 252

582 Index

Kelvin transform, 45, 47, 235, 259kernel– (elliptic) Dirichlet Green, 5, 440, 454– Dirichlet heat, 5, 78, 83, 86, 136,

219, 274, 440, 444, 453, 454, 459– Gaussian heat, 5, 101, 102, 129, 131,

139, 289, 335, 360, 440, 460Kranoselskii genus, 32

Lagrange multiplier, 20Liouville-type theorem– elliptic equation, 36, 40, 65, 206– elliptic inequality, 37– elliptic system, 260, 266– parabolic equation, 150, 210, 339Ljusternik-Schnirelman, 26localization of singularities, 64, 271localization of trajectories, 210localized nonlinearity, 393, 402Lyapunov functional, 293, 529– strict, 312, 359, 530

mass– dissipation of, 288– growth of, 360matched asymptotics, 247, 360maximal regularity, 163, 165, 236, 470maximum principle– elliptic, 507– nonlocal, 344, 523– parabolic, 509, 512– strong (elliptic), 507– strong (parabolic), 511– systems, 522– very weak (elliptic), 447– very weak (parabolic), 450, 515memory term, 421

minimax methods, 29minimization, 236model problem– elliptic, ix– parabolic, xmollifier, 431, 433, 509monotonicity– of solutions in time, 174, 180, 191,

219, 283, 340, 424, 520, 531– via moving planes, 38, 49, 157, 261Moser-type iteration, 90, 216mountain pass– energy, 117– theorem, 29moving planes– elliptic, 38, 45, 47, 49, 67, 68, 70,

253, 261– parabolic, 157, 193, 235multiplier argument (or technique),

533

Nehari functional, 116Nemytskii mapping, 33, 474Newton potential, see fundamental

solutionnonlinear boundary conditions, x, 225,

230, 469nonuniqueness, 76, 77, 143, 230, 272

Ohmic heating, 412operator– realization of, 2optimal controls, 236

Index 583

Palais-Smale sequence, 29, 33parabolic boundary, 1peaking solution, 228perturbation– of the model problem (elliptic), 35– of the model problem (parabolic),

313, 319, 330, 338, 348– singular, 297population genetics, 108potential well, 116

quenching, 100

radial– function, 2– monotonicity, 518– nonincreasing function, 2realization of an operator, 2reflection, 45removable singularity, 13rescaling method– elliptic, 36, 40, 65, 260, 263– in similarity variables, 179, 183– parabolic, 162, 206, 212, 215, 286,

339

scaling, 99, 327, 330, 391– invariance, 116scaling exponents, 252, 272Schauder estimates– elliptic, 430– parabolic, 438Schwarz symmetrization, 22self-adjoint operator, 104, 108, 312,

434, 439, 450, 456, 488, 491self-similar– asymptotically, 142– blow-up behavior, 189, 195, 347

– solution (backward), 167, 169, 197,302, 342, 353, 355

– solution (forward), 77, 133, 141–143,216, 330, 363

– subsolution, 321, 419– supersolution, 131, 335, 420semigroup– analytic, 466– Dirichlet heat, 439separation lemma, 96similarity variables– backward, 180, 195, 341– forward, 104smoothing estimate, smoothing effect,

76Sobolev hyperbola, 253, 260solution– L1, L1

δ, very weak (elliptic), 8– classical (elliptic), 7– classical (parabolic), 75– distributional (elliptic), 8– distributional (parabolic), 101– integral (parabolic), 77, 78, 79, 86,

227, 443– periodic, 234– singular (elliptic), 11, 50, 62, 131,

168, 185, 187, 267, 271, 364, 458– strong (elliptic), 58, 65, 429– strong (parabolic), 438, 473, 511– variational (elliptic), 8, 56– weak, weak-L1

δ (parabolic), 78, 443stability– of equilibria, 112, 113, 123, 131, 142,

308, 378, 485, 532– of self-similar solutions, 142stabilization, 123Stampacchia method, 508, 512standard function, 387– sub-, 387

584 Index

– super-, 387starshaped domain, 18, 19, 22, 26,

123, 161, 237, 253subsolution, supersolution, 507system– cooperative, 251, 522– Gierer-Meinhardt, 297– Lane-Emden, 251– with mass-dissipation, 288

test-function, 40, 203– Gaussian, 101, 281– rescaled, 37, 102, 262, 330, 464– singular, 59, 205– torsion, 17, 124, 223, 280, 359, 379,

397, 447test-function argument (or technique),

533thermistor, 413threshold solution, threshold trajec-

tory, 171, 177, 228, 237, 239, 359topology of the domain, 26trajectory, 528transition– from global existence to blow-up,

237– from single-point to global blow-up,

398

uniformly local spaces, 3, 86, 217, 460uniqueness– elliptic, 22, 25– local (parabolic), 76, 87, 272, 398,

471, 495universal bound, 193, 202, 230, 234,

280, 338, 359, 395

variation-of-constants formula, 77,443, 446, 468

variational– identity, see identity (Pohozaev,

Rellich-Pohozaev type)– methods, 20– solution, see solution– structure, 55, 251, 258, 339, 411viscous Hamilton-Jacobi equation,

314, 355

weighted spaces, see also Lpδ-spaces

– Lebesgue, 104, 434, 491– Sobolev, 104, 137, 239, 434, 491well-posedness, 75, 89, 190, 230, 273,

314, 355, 394, 470, 495

zero number, 151, 168, 172, 185, 244,526

superlinear parabolic problems: blow-up, global existence and …kaplicky/pages/pages/... · 2013....

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