Pullback attracting inertialmanifolds for nonautonomous

dynamical systemsby

Norbert Koksch and Stefan Siegmund

CDSNS 2001-379

Pullback Attracting Inertial Manifolds forNonautonomous Dynamical Systems

Norbert Koksch∗ and Stefan Siegmund†

Abstract. In this paper we present an abstract approach to inertial manifolds for nonautonomousdynamical systems. Our result on the existence of inertial manifolds requires only two geomet-rical assumptions, called cone invariance and squeezing property, and two additional technicalassumptions, called boundedness and coercivity property. Moreover we give conditions whichensure that the global pullback attractor is contained in the inertial manifold.In the second part of the paper we consider special nonautonomous dynamical systems, namelytwo-parameter semi-flows. As a first application of our abstract approach and for reason of com-parison with known results we verify the assumptions for semilinear nonautonomous evolutionequations whose linear part satisfies an exponential dichotomy condition and whose nonlinearpart is globally bounded and globally Lipschitz.

Keywords and phrases:Inertial manifolds, pullback attractor, nonautonomous dynamical sys-tems

Mathematics subject classification 2000:34C30, 34D45, 34G20, 35B42, 37L25∗Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germanye-mail: koksch@math.tu-dresden.de

†Georgia Institute of Technology, CDSNS, Atlanta, GA 30332-190, USAe-mail: siegmund@math.gatech.edu

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1 Introduction

Let us consider a dissipative nonlinear evolution equation of the form

u+Au= f (u)

in a Banach spaceX, whereA is a linear sectorial operator with compact resolvent andf is anonlinear function. Such an evolution equation may be an ordinary differential equation (X =Rn)or the abstract formulation of a semilinear parabolic differential equation withX as a suitablefunction space over the spatial domain. In the last case,A corresponds to a linear differentialoperator andf is a nonlinearity which may involve derivatives of lower order thanA.

The dissipativity finds its expression in the fact that there is a bounded setA in X which absorbsbounded subsets of the phase space after a finite time. Given a compact setA which is absorbingand positively invariant with respect to the semi-flow, theω-limit set of A under the semi-flowhas some interesting properties: There is a compact, invariant setA in A which attracts alltrajectories. Such a setA is called theglobal attractor[Tem97], and it is maximal in the sensethat it is the largest set with this property. Often global attractors have a finite (fractal) dimensionand, therefore, they are important objects for the study of long time behavior. At the present levelof understanding of dynamical systems, the attractors are expected to be very complicated objects(fractals) and their practical utilization, for instance for numerical simulations, may be difficult.Because of the finite dimensionality of the attractor, it is a natural question if the attractor couldbe generated by a (possibly high dimensional) system of ordinary differential equations, i.e. ifthe flow on the attractor could be generated by a system of ordinary differential equations. Anindirect way to obtain such a system is to embed the attractorA into a smooth finite dimensionalmanifold.

Inertial manifolds are positively invariant, exponentially attracting, finite dimensional Lipschitzmanifolds. They go back to D. Henry and X. Mora [Hen81, Mor83] and were first intro-duced and studied by P. Constantin, C. Foias, B. Nicoalenko, G.R. Sell and R. Temam [FST85,FNST85, CFNT86] for selfadjointA. For the construction of inertial manifolds withA beingnon-selfadjoint see for example [SY92] and [Tem97]. Inertial manifolds are generalizations ofcenter-unstable manifolds and they are more convenient objects which capture the long-time be-havior of dynamical systems. If such a manifold exists, then it contains the global attractorA.Usually an inertial manifoldM is seeked as the graph of a sufficiently smooth functionmonPX,i.e.

M = graph(m) := {x+m(x) : x∈ PX} ,

whereP is a finite dimensional projector. The finite dimensionality and the exponential attractingproperty permit the reduction of the dynamics of the infinite or high dimensional equation to thedynamics of a finite or low dimensional ordinary differential equation

x+Ax= P f(x+m(x)) in PX

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called inertial form system. A stronger reduction property is the asymptotical completenessproperty [Rob96]: Each trajectory of the evolution equation tends exponentially to a trajectoryin the inertial manifold.

Thus, one can reduce (in some sense) an infinite-dimensional physical system to a system with afinite number of degrees of freedom and use known properties of ordinary differential equationsin Rn for the qualitative analysis of the behavior of the solutions of the original equation for largetime. Another advantage of studying the inertial form system is the reduction of the dimensionof the original problem which could be important for the numerical analysis. In concrete appli-cations, theN-dimensional projectorP is the projector onto the eigenspace spanned by the firstNFourier modes ofL, i.e. by eigenvectors ofA belonging to the set of theN smallest eigenvalues ofA. Hence the inertial form systems describes the dynamics of the slow modes. For applicationsof the concept of inertial manifolds see for example [RB95, SK95, BH96, MSZ00].

There are a few ways of constructing an inertial manifold. Most of them are generalizationof methods developed for the construction of unstable, center-unstable or center manifolds forordinary differential equations. Following the introduction of [Wig94], there are at least thefollowing two basic methods:

The Lyapunov-Perron method has been developed by A.M. Lyapunov [Lya47, Lya92] and O. Per-ron [Per28, Per29, Per30] for the proof of the existence of stable and unstable manifolds of hy-perbolic equilibrium points. In the context of ordinary differential equations, it deals with theintegral equation formulation of the differential equation and constructs the invariant manifoldas a fixed point of an operator that is derived from this integral equation. This method hasbeen used in many different situations. The book of J. Hale [Hal80] may serve as a good ref-erence. In the infinite dimensional setting, the method is used by D. Henry [Hen81] to provethe existence of stable, unstable and center manifolds for semilinear parabolic equations. In[FST88, Tem88, CFNT89b, FST89, DG91], it has been adapted for the construction of inertialmanifolds. At the moment, the Lyapunov-Perron method is the most common method for inertialmanifolds.

Hadamard’s method [Had01], also called graph transformation method, has been developed toprove the existence of stable and unstable manifolds of fixed points of diffeomorphisms. Thegraph transformation method is more geometrical in nature than the Lyapunov-Perron method.In presence of a hyperbolic fixed point, the stable and unstable manifolds are constructed asgraphs over the linearized stable and unstable subspaces. For the infinite dimensional case, werefer to [BJ89]. A review on different construction methods (including Hadamard’s method) forinertial manifolds can be found in [LS89, Nin93].

The above mentioned notion of inertial manifolds is translated and extended to more generalclasses of differential equations like nonautonomous differential equations, [GV97, WF97, LL99],retarded parabolic differential equations, [TY94, BdMCR98], or differential equations with ran-dom or stochastic perturbations, [Chu95, BF95, CL99, CS01, DLS01].

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The construction of inertial manifolds often is redone for different classes of equations. Ouraim is to separate the general structure of the construction from the technical estimates whichvary from example to example. To achieve this goal we develop the existence result of inertialmanifolds on the abstract level of nonautonomous dynamical systems and then apply it to explicitnonautonomous evolution equations under various assumptions.

In Sec.2 we introduce the abstract notion of a nonautonomous dynamical system extendingdynamical systems to the nonautonomous situation. For a nonautonomous dynamical systemwe introduce inertial manifolds and gloabl pullback attractors. We formulate a cone invariance,squeezing, boundedness and coercivity property which are assumed in Subsection3.4where theexistence of an inertial manifold is proved. Moreover we give conditions which imply that theglobal pullback attractor is contained in the inertial manifold. The assumptions of the existenceresult are formulated on the general level of nonautonomous dynamical systems and are inde-pendent of a specific equation which generates the nonautonomous dynamical system.

In Sec.3 we apply it to special nonautonomous dynamical systems, namely two-parameter semi-flows. As a first application we verify the assumptions for nonautonomous semilinear evolutionequations whose linear part satisfies an exponential dichotomy condition and whose nonlinearpart is globally bounded and globally Lipschitz under the assumption that a special algebraicproblem is solvable. For this we develop and apply comparison theorems. The algebraic problemis solvable if the exponential rates in the exponential dichotomy condition satisfy a gap conditionwhose form depends on the used Lipschitz inequality and the concrete form of the exponentialdichotomy condition. Finally we compare our results with known ones.

Applications to nonautonomous dynamical systems which are generated by stochastic or retardedpartial differential equations are subject of a forthcoming paper.

2 Nonautonomous Dynamical Systems

2.1 Preliminaries

Let (X,‖ · ‖) be a Banach space.

Definition 2.1 (Nonautonomous Dynamical System (NDS)).A nonautonomous dynamicalsystem(NDS) onX is acocycleϕ over adriving systemθ on a setB, i.e.

(i) θ : R×B→B is adynamical system, i.e. the familyθ(t, ·) = θ(t) : B→B of self-mappingsof B satisfies thegroup property

θ(0) = idB , θ(t +s) = θ(t)◦θ(s)

for all t,s∈ R.

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(ii) ϕ :R≥0×B×X→X is acocycle, i.e. the familyϕ(t,b, ·) = ϕ(t,b) : X→X of self-mappingsof X satisfies thecocycle property

ϕ(0,b) = idX , ϕ(t +s,b) = ϕ(t,θ(s)b)◦ϕ(s,b)

for all t,s≥ 0 andb∈B. Moreover(t,x) 7→ ϕ(t,b,x) is continuous.

Remark2.2. (i) The setB is calledbaseand in applications it has additional structure, e.g. it is aprobability space, a topological space or a compact group and the driving system has additionalregularity, e.g. it is ergodic or continuous.

(ii) The pair of mappings

(θ,ϕ) : R≥0×B×X→B×X , (t,b,x) 7→ (θ(t,b),ϕ(t,b,x))

is a special semi-dynamical system a so-calledskew product flow(usually one requires addi-tionally that (θ,ϕ) is continuous). IfB = {b} consists of one point then the cocycleϕ is asemi-dynamical system.

(iii) We use the abbreviationsθtb or θ(t)b for θ(t,b) andϕ(t,b)x for ϕ(t,b,x). We also say thatϕ is an NDS to abbreviate the situation of Definition2.1.

Definition 2.3 (Nonautonomous Set).A family M = (M(b))b∈B of non-empty setsM(b)⊂ X

is called anonautonomous setandM(b) is called theb-fiberof M or thefiber of M overb. Wesay thatM is closed, open, bounded, or compact, if every fiber has the corresponding property.For notational convenience we use the identificationM' {(b,x) : b∈B, x∈M(b)} ⊂B×X.

Definition 2.4 (Invariance of Nonautonomous Set).A nonautonomous setM is calledforwardinvariant under the NDSϕ, if ϕ(t,b)M(b)⊂M(θtb) for t ≥ 0 andb∈ B. It is calledinvariant,if ϕ(t,b)M(b) = M(θtb) for t ≥ 0 andb∈B.

Definition 2.5 (Inertial Manifold). Let ϕ be an NDS. Then a nonautonomous setM is called(nonautonomous)inertial manifoldif

(i) every fiberM(b) is afinite-dimensional Lipschitz manifoldin X of dimensionN for anN ∈N;

(ii) M is invariant;

(iii) M is exponentially attracting, i.e. there exists a positive constantη such that for everyb∈B

andx∈M(b) there exists anx′ ∈M(b) with

‖ϕ(t,b)x−ϕ(t,b)x′‖ ≤ Ke−ηt for t ≥ 0 andb∈B

and a constantK = K(b,x,x′)> 0.

The property (iii) is also calledexponential tracking propertyor asymptotic completeness prop-ertyandx′ or ϕ(·,b)x′ is said to be theasymptotic phaseof x or ϕ(·,b)x, respectively.

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Recall that ifD andA are nonempty closed sets inX, the Hausdorff semi-metricd(D|A) isdefined by

d(D|A) := supx∈D

d(x,A) , d(x,A) := infy∈A

d(x,y) = infy∈A‖x−y‖ .

Now we want to generalize the attractor notion of dynamical systems to nonautonomous dynam-ical systems. A natural generalization of convergence to a nonautonomous setA seems to be theforwards running convergencedefined by

d(ϕ(t,b)x,A(θtb))→ 0 for t→ ∞ .

However, this does not ensure convergence to a specific component setA(b) for a fixedb. Forthat one needs to start “progressively earlier” atθ−tb in order to “finish” atb. This leads to theconcept ofpullback convergencedefined by

d(ϕ(t,θ−tb)x,A(b))→ 0 for t→ ∞ .

Using this we can define apullback attractor.

Definition 2.6 (Pullback Attracting, Absorbing, Attractor). Let ϕ be an NDS andA be anonautonomous set.

(i) A is called(globally) pullback attractingif for every bounded setD⊂ X andb∈B

limt→∞

d(ϕ(t,θ−tb)D|A(b)) = 0.

(ii) A is called(globally) pullback absorbingif A is bounded and for every bounded setD⊂ X

andb∈B there is aT > 0 with

ϕ(t,θ−tb)D⊂A(b) for all t ≥ T .

(iii) A is called(global) pullback attractorof ϕ if A is compact, invariant and pullback attracting.

The concept of pullback convergence was introduced in the mid 1990s in the context of ran-dom dynamical systems (see Crauel and Flandoli [CF94], Flandoli and Schmalfuss [FS96], andSchmalfuss [Sch92]) and has been used e.g. in numerical dynamics. Note that a similar ideahad already been used in the 1960s by Mark Krasnoselski [Kra68] to establish the existence ofsolutions that exist and remain bounded on the entire time set.

Now we define a handy notion (see Ludwig Arnold [Arn98, Definition 4.1.1(ii)]) excluding ex-ponential growth of a function.

Definition 2.7 (Temperedness).A function R : B→ ]0,∞[ is calledtempered from aboveif foreveryb∈B

limsupt→±∞

1|t|

logR(θtb) = 0.

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Note that the following characterization holds.

Corollary 2.8. Suppose that R: B→ ]0,∞[ is a nonautonomous variable. Then the followingstatements are equivalent:

(i) R is tempered from above.

(ii) For everyε> 0 and b∈B there exists a T> 0 such that

R(θtb)≤ eε|t| for |t| ≥ T .

Definition 2.9 (Nonautonomous Projector).Let ϕ be an NDS. A familyπ = (π(b))b∈B ofprojectorsπ(b) ∈ L(X,X) in X is callednonautonomous projector.

(i) π is calledtempered from aboveif b 7→ ‖π(b)‖ is tempered from above.

(ii) π is calledN-dimensionalfor anN ∈ N if dimimπ(b) = N for everyb∈B.

2.2 Inertial Manifolds for Nonautonomous Dynamical Systems

Let π1 be anN-dimensional nonautonomous projector inX. We define thecomplementary pro-jector

π2(b) := idX−π1(b) for b∈B .

ThenX1(b) := π1(b)X and X2(b) := π2(b)X , b∈B

define nonautonomous setsXi consisting of complementary linear subspacesXi(b) of X, i.e.X1(b)⊕X2(b) = X. For this fact we also writeX1⊕X2 = B×X.

We want to construct a nonautonomous inertial manifold

M = (M(b))b∈B

consisting of manifoldsM(b) which are trivial in the sense that each of them can be describedby a single chart, i.e.

M(b) = graph(m(b, ·)) := {x1 +m(b,x1) : x1 ∈ X1(b)}

with m(b, ·) = m(b) : X1(b)→ X2(b).

For a positive constantL we introduce the nonautonomous set

CL := {(b,x) ∈B×X : ‖π2(b)x‖ ≤ L‖π1(b)x‖} .

Since the fibersCL(b) are cones it is called(nonautonomous) cone.

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Definition 2.10 (Cone Invariance).The NDSϕ satisfies the (nonautonomous)cone invariancepropertyfor a coneCL if there is aT0≥ 0 such that forb∈B andx,y∈ X,

x−y∈ CL(b)

impliesϕ(t,b)x−ϕ(t,b)y∈ CL(θtb) for t ≥ T0 .

Now we define a property of a cocycleϕ which describes a kind of squeezing outside a givencone.

Definition 2.11 (Squeezing Property).The NDSϕ satisfies the (nonautonomous)squeezingpropertyfor a coneCL if there exist positive constantsK1, K2 andη such that for everyb∈ B,x,y∈ X andT > 0 the identity

π1(θTb)ϕ(T,b)x = π1(θTb)ϕ(T,b)y

implies for allx′ ∈ X with π1(b)x′ = π1(b)x andx′−y∈ CL(b) the estimates

‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)y]‖ ≤ Kie−ηt‖π2(b)[x−x′]‖ , i = 1,2,

for t ∈ [0,T].

Remark2.12. We consider the special case thatB = {b} is a singleton. Then the NDSϕ definesa semiflowSon X by Stx = ϕ(t,b)x for (t,x) ∈ R≥0×X and we may replaceπi(b) by πi . As aspecial case of the cone invariance property we obtain the

Autonomous Cone Invariance Property:There existsT0≥ 0 such thatx,y∈ X with

‖π2[x−y]‖ ≤ L‖π1[x−y]‖

implies‖π2[Stx−Sty]‖ ≤ L‖π1[Stx−Sty]‖ for t ≥ T0 .

The squeezing property in the autonomous case becomes the following

Autonomous Squeezing Property:There exist positive constantsK1, K2, η such thatfor everyx,y∈ X andT > 0 the identity

π1STx = π1STy

implies for allx′ ∈ X with π1x′ = π1x and‖π2[x′−y]‖ ≤ L‖π1[x′−y]‖ the estimates

‖πi [Stx−Sty]‖ ≤ Kie−ηt‖π2[x−x′]‖ , i = 1,2,

for t ∈ [0,T].

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A combination of cone invariance and squeezing properties for evolution equations, sometimescalled strong squeezing property, was first introduced for the Kuramoto-Sivashinsky equations in[FNST85, FNST88], an abstract version of it was developed in [FST89], another formulation ofit can be found for example in [Tem88, FST88, CFNT89a, Rob93, JT96]. Essentially, a strongsqueezing property states that if the difference of two solutions of the evolution equation belongsto a special cone then it remains in the cone for all further times (that is the cone invariance prop-erty); otherwise the distance between the solutions decays exponentially (that is the squeezingproperty).

Our autonomous cone invariance property withT0 = 0 corresponds to these cone invariance prop-erties. Our autonomous squeezing property is a modification of the usual squeezing properties,however, at least for semilinear parabolic evolution equations, a strong squeezing property andan additional cone invariance property imply our autonomous squeezing property.

Definition 2.13 (Boundedness Property).The NDSϕ satisfies the (nonautonomous)bounded-ness propertyif for all t ≥ 0, b∈ B and allM1 ≥ 0 there exists anM2 ≥ 0 such that forx∈ X

with ‖π2(b)x‖ ≤M1 the estimate

‖π2(θtb)ϕ(t,b)x‖ ≤M2

holds.

Definition 2.14 (Coercivity Property). The NDSϕ satisfies the (nonautonomous)coercivityproperty if for all t ≥ 0, b∈ B and allM3 ≥ 0 there exists anM4 ≥ 0 such that forx∈ X with‖π1(b)x‖ ≥M4 the estimate

‖π1(θtb)ϕ(t,b)x‖ ≥M3

holds.

The coercivity property will ensure the existence of global homeomorphisms used for the defi-nition of the graph transformation mapping. With the boundedness property we will ensure thatthe graph transformation mapping maps bounded graphs on bounded graphs.

Remark2.15. As we will show later in Sec.3.4, for evolution equations the coercivity andboundedness property ofϕ follows from the boundedness of the nonlinearity and exponentialdichotomy properties of the linear part.

We say thatϕ satisfies aninverse Lipschitz inequalityif for all t ≥ 0, b∈ B there exists aK > 0such that

‖π1(b)x−π1(b)y‖ ≤ K‖π1(θtb)ϕ(t,b)x−π1(θtb)ϕ(t,b)y‖

holds forx,y∈ X. Obviously, the inverse Lipschitz property ofϕ implies the coercivity propertyof ϕ. For evolution equations, the inverse Lipschitz property follows from a uniform Lipschitzproperty of the nonlinearity and exponential dichotomy properties of the linear part.

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Theorem 2.16 (Existence of Inertial Manifold). Let ϕ be an NDS on a Banach spaceX overa driving systemθ : R×B→ B on a setB and assume thatϕ satisfies the cone invariance,squeezing, coercivity and boundedness property. Then there exists an inertial manifoldM =(M(b))b∈B of ϕ with the following properties:

(i) M(b) is a graph inX1(b)⊕X2(b),

M(b) = {x1 +m(b,x1) : x1 ∈ X1(b)}

with a mapping m(b, ·) = m(b) : X1(b)→ X2(b) which is globally Lipschitz continuous

‖m(b,x1)−m(b,y1)‖ ≤ L‖x1−y1‖ .

(ii) M is exponentially attracting, more precisely

‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)x′]‖ ≤ Kie−ηt‖π2(b)x−m(b,π1(b)x)‖

for t ≥ 0, i = 1,2, with an asymptotic phase x′= x′(b,x)∈M(b) of x and the constants K1,K2> 0from the squeezing property which are independent of b and x.

(iii) If in addition π1 is tempered from above, thenM is pullback attracting, more precisely, thereis a T≥ 0 such that for each bounded setD⊂ X

d(ϕ(t,θ−tb)D|M(b))≤ e−ηt/2d(D|M(θ−tb)) for t ≥ T . (1)

Proof. As the proof is rather involved we split it into four parts. First we define the graphtransformation mapping. In the second step we show that it has a unique fixed pointm(b) whichgives rise to a nonautonomous invariant setM of Lipschitz manifoldsM(b). In the third stepthe exponential tracking property is proved and in the fourth step we investigate the pullbackattractivity ofM.

Step 1: Definition of graph transformation mapping

We construct the manifoldsM(b) = graph(m(b)) as the fixed point of the cocycleϕ acting on acertain class of functionsg with

g(b, ·) = g(b) : X1(b)→ X2(b) , b∈B .

The setG of mappings of the form

X1 3 (b,x1) 7→ (b,g(b,x1)) ∈ X2 ,

such thatg is bounded andg(b, ·) is continuous for everyb∈B is a Banach space with the norm

‖g‖G = sup(b,x1)∈X1

‖g(b,x1)‖ .

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X1(b)

θt(b)

X2(θt(b))

X1(θt(b))

B

X2(b)

b

ϕ(T,b)ϕ(T,b)graph(g(θt(b)))

=: graph((GTg)(θTb, ·)

)

graph(g(b)))

Figure 1: Graph transformation mappingGT

Moreover letGL denote the subset ofG containing all mappings which satisfy the global Lipschitzcondition

‖g(b,x1)−g(b,y1)‖ ≤ L‖x1−y1‖

for (b,x1),(b,y1) ∈ X1 with L from the cone invariance property. Note that bothG andGL ⊂ G

are complete metric spaces.

Let T > 0 be arbitrary, but fixed. We wish to define thegraph transformation mapping GT : GL→G such that

graph((GTg)(θTb, ·)) = ϕ(T,b)graph(g(b, ·))

(see Fig.2.2) and this means that ˜x∈graph((GTg)(θTb, ·)) equalsϕ(T,b)x for somex∈graph(g(b, ·)).More precisely, for an ˜x1 ∈ X1(θTb) we want to define(GTg)(θTb, x1) = x2 ∈ X2(θTb) by de-termining anx∈ graph(g(b, ·)) such that

π1(θTb)ϕ(T,b)x = x1 and π2(θTb)ϕ(T,b)x =: (GTg)(θTb, x1) .

To this end we show that for eachT > 0, b ∈ B, x1 ∈ X1(θTb), g ∈ GL, the boundary valueproblem

x∈ graph(g(b, ·)) , π1(θTb)ϕ(T,b)x = x1 (2)

has a unique solutionx = β(T,b, x1,g).

Uniquenessof a solution of (2). Assume there existx andy with

x,y∈ graph(g(b, ·)) , π1(θTb)ϕ(T,b)x = π1(θTb)ϕ(T,b)y = x1 .

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We getx−y∈ CL(b) and the squeezing property (withx′ = x) implies the identity

ϕ(t,b)x = ϕ(t,b)y for t ∈ [0,T] .

Hencex = y and there is at most one solutionx = β(T,b, x1,g) of (2).

Existenceof a solution of (2). LetT > 0,b∈B, g∈ GL be fixed and defineH : X1(b)→X1(θTb)by

H(x1) := π1(θTb)ϕ(T,b)(x1 +g(b,x1)) .

By the continuity ofϕ(T,b) andg(b, ·), H is continuous. For ˜x1 ∈ HX1(b) ⊂ X1(θTb), anyx1

in the preimageH−1(x1) = {x1 ∈ X1(b) : H(x1) = x1} gives rise to a solutionx = x1 + g(b,x1)of the boundary value problem (2). As we have already seen, there exists at most one solutiondenoted byβ(T,b, x1,g) and therefore the inverseH−1 of H is given by

H−1(x1) = π1(b)β(T,b, x1,g) on HX1(θTb) .

Now we show the continuity ofH−1 : HX1(b)→ X1(b). Suppose, there is aξ ∈ HX1(b) ⊂X1(θTb) such thatH−1 is not continuous atξ. Then there areε > 0 and a sequence(ξk)k∈N inX1(θTb) such thatξk→ ξ0 ask→ ∞ and

‖ξk−ξ0‖ ≥ ε for all k∈ N (3)

whereξk := π1(b)β(T,b, ξk,g) for k = 0,1, . . ..

First, we suppose that there is a subsequence of(ξk)k∈N, denoted for shortness again by(ξk)k∈N,with ‖ξk‖→ ∞ ask→ ∞. Then the coercivity property ofϕ implies

‖ξk‖= ‖H(ξk)‖= ‖π1(θTb)ϕ(T,b)(ξk +g(b,ξk))‖→ ∞ ask→ ∞ ,

but this contradictsξk→ ξ0.

Therefore we have proved that(ξk)k∈N is bounded. SinceX1(b) is a finite-dimensional space,there is a convergent subsequence, denoted for shortness again by(ξk)k∈N, with a limit

ξ∞ = limk→∞

ξk ∈ X1(b) . (4)

By the continuity ofH, we haveH(ξk)→ H(ξ∞). Since alsoH(ξk) = ξk→ ξ0 = H(ξ0) we getξ0 = ξ∞ in contradiction to (3) and (4). Therefore,H andH−1 are continuous.

Now we show thatH is onto, i.e. satisfies

HX1(b) = X1(θTb) . (5)

By the coercivity ofϕ, we have the norm coercivity‖H(ξ)‖→∞ for ‖ξ‖→∞ of H. SinceX1(b)is finite-dimensional,H is a sequentially compact mapping. By [Rhe69, Theorem 3.7],H is ahomeomorphism fromX1(b) ontoX1(θTb) and hence we have (5).

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Thus we have unique solvability of (2), and we can define the graph transformation mappingGT

by(GTg)(θTb, x1) = π2(θTb)ϕ(T,b)β(T,b, x1,g)

for T > 0, b∈B, x1 ∈ X1(θTb), andg∈ GL. Note that we have

graph((GTg)(θTb, ·)) = ϕ(T,b)graph(g(b, ·)) .

SinceH−1, g(b, ·) andx 7→ π1(b)ϕ(T,b)x are continuous, and

β(T,b, x1,g) = H−1(x1)+g(b,H−1(x1))

holds, the mapping(GTg)(θTb, ·) is also continuous.

It remains to show that‖GTg‖G < ∞ .

Sinceg∈ G there is aM1 with ‖π2(b)x‖ ≤ M1 for all b∈ B and allx∈ graph(g(b, ·)). By theboundedness property ofϕ there is aM2 with ‖π2(θTb)ϕ(t,b)x‖ ≤M2 for all b∈B and allx∈X

with ‖π2(b)x‖ ≤M1. Let b∈B0, x1 ∈X1(θTb) be arbitrary. Thenβ(T,b, x1,g) ∈ graph(g(b, ·)),hence‖π2(b)β(T,b, x1,g)‖ ≤M1, and therefore

‖(GTg)(θTb, x1)‖= ‖π2(θTb)ϕ(T,b)β(T,b, x1,g)‖ ≤M2

proving that‖GTg‖G < ∞, i.e.GTg∈ G.

Step 2: Unique fixed-point of graph transformation mappings

Let T > 0, b ∈ B, x1, x2 ∈ X1(θTb), g ∈ GL be arbitrary. Sinceβ(T,b, x1,g), β(T,b, x2,g) ∈graph(g(b, ·)), we get

β(T,b, x1,g)−β(T,b, x2,g) ∈ CL(b)

and the cone invariance property implies forT ≥ T0 a Lipschitz estimate forGTg,

‖(GTg)(θTb, x1)− (GTg)(θTb, x2)‖≤ L‖π1(θTb)(ϕ(T,b)β(T,b, x1,g)−ϕ(T,b)β(T,b, x2,g))‖= L‖x1− x2‖ ,

i.e.GT mapsGL into G for everyT ≥ 0, and it is self-mapping forT ≥ T0.

Now letT ≥ 0, b∈B, x1 ∈ X1(θTb), g,h∈ G, andx∈ graph(g(b, ·)), y∈ graph(h(b, ·)) with

π1(θTb)ϕ(T,b)x = π1(θTb)ϕ(T,b)y = x1 .

Definex′ := π1(b)x+h(b,π1(b)x) .

13

Thenx′−y∈ CL, and the squeezing property implies

‖π2(θTb)[ϕ(T,b)x−ϕ(T,b)y]‖ ≤ K2e−ηT‖g(b,π1(b)x)−h(b,π1(b)x)‖

for T ≥ 0. Thus

‖(GTg)(θTb, x1)− (GTh)(θTb, x1)‖≤ K2e−ηT‖g(b,π1(b)β(T,b, x1,g))−h(b,π1(b)β(T,b, x1,g))‖ ,

and passing to the sup over all(θTb, x1) ∈ X1 we get

‖GTg−GTh‖G ≤ κ(T)‖g−h‖G

for all T > 0, g,h∈ GL, whereκ(T) := K2e−ηT .

Sinceη > 0, there is a positiveT1 ≥ T0 with κ(T) < 1 for T ≥ T1. Thus, forT ≥ T1, GT isa contractive self-mapping on the complete metric spaceGL. Now choose and fix an arbitraryT ≥ T1 and letm denote the unique fixed-point ofGT in GL. We show thatm is the uniquefixed-point ofGT for everyT ≥ 0.

For everyT ≥ 0 the mappingGTm∈ G is uniquely determined by the graphs

graph((GTm)(θTb, ·)) = ϕ(T,b)graph(m(b, ·)) , b∈B .

For T ≥ 0 andb∈B we have the identity

graph((GT+Tm)(θT+Tb, ·)) = ϕ(T + T,b)graph(m(b, ·))= ϕ(T,θTb)ϕ(T,b)graph(m(b, ·))= ϕ(T,θTb)graph(m(θTb, ·))= graph((GTm)(θT+Tb, ·))

and thereforeGTm= GT+Tm∈ GL. Hence the compositionGTGT ′m makes sense forT,T ′ ≥ 0and we get

graph((GTGT ′m)(θT+T ′b, ·)) = ϕ(T,θT ′b)graph(GT ′m(θT ′b, ·))= ϕ(T,θT ′b)ϕ(T ′,b)graph(m(b, ·))= ϕ(T +T ′,b)graph(m(b, ·))= graph((GT+T ′m)(θT+T ′b, ·))

and thereforeGTGT ′m= GT ′GTm= GT+T ′m for T,T ′ ≥ 0. We get

GT(GTm) = GT(GTm) = GTm.

14

ThusGTm equals the unique fixed-pointm of GT and we have

GTm= m for T ≥ 0.

To prove the uniqueness of the fixed-pointm of GT , assume thatm∗ is another fixed-point. Butthenm andm∗ are both fixed-points ofGkT for everyk∈ N. Choosingk large enough such thatkT ≥ T1 we know thatGkT has a unique fixed-point and this impliesm= m∗.

Thusm is the unique mapping inGL with the invariance property

ϕ(t,b)graph(m(b, ·)) = graph(m(θtb, ·)) for t ≥ 0 andb∈B .

We defineM(b) := graph(m(b, ·)) for b∈B.

Step 3: Existence of asymptotic phase

Let b∈B andx∈X be arbitrary and let(tk)k∈N be a monotonously increasing sequence of posi-tive real numberstk with tk→ ∞ for k→ ∞. Definey′ := π1(b)x+m(b,π1(b)x) ∈ graph(m(b, ·))and

xk := β(tk,b,π1(θtkb)ϕ(tk,b)x,m) ∈ graph(m(b, ·)) .

We gety′−xk ∈ CL(b) and the squeezing property implies fori = 1,2, t ∈ [0, tk]

‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)xk]‖ ≤ Kie−ηt‖π2(b)x−m(b,π1(b)x)‖ .

In particular, we find fori = 1 andt = 0

‖π1(b)xk‖ ≤ ‖π1(b)x‖+‖π1(b)[x−xk]‖≤ ‖π1(b)x‖+K1‖π2(b)x−m(b,π1(b)x)‖ .

Therefore we have proved that(π1(b)xk)k∈N ⊂ X1(b) is bounded. SinceX1(b) is a finite-dimensional space, there is a convergent subsequence, denoted again by(π1(b)xk)k∈N. Since

xk = π1(b)xk +m(b,π1(b)xk)

andm(b, ·) is continuous, also(xk)k∈N is converging to some

x′ ∈ graph(m(b, ·)) ,

Then we get fori = 1,2

‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)x′]‖≤ ‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)xk]‖+‖πi(θtb)[ϕ(t,b)xk−ϕ(t,b)x′]‖≤ Kie

−ηt‖π2(b)x−m(b,π1(b)x)‖+‖πi(θtb)[ϕ(t,b)xk−ϕ(t,b)x′]‖

15

for all T > 0, t ∈ [0,T] and allk∈ N>0 with tk ≥ T. By the continuity ofϕ(t,b), and because ofxk→ x′, the second term can be made arbitrary small for each fixedt ∈ [0,T] choosingk largeenough. Therefore,

‖πi(θtb)[ϕ(t,b)x−ϕ(t,b)x′]‖ ≤ Kie−ηt‖π2(b)x−m(b,π1(b)x)‖

for t ≥ 0, i.e.x′ ∈M(b) is an asymptotic phase ofx.

Step 4: Pullback attractivity

Note that withπ1 also the complementary projectorπ2 is tempered from above. Sinceϕ(t,b)x′ ∈M(θtb) for everyx∈ X, t ≥ 0, b∈B, Step 3 implies

d(ϕ(t,b)x,M(θtb))≤ Ke−ηt‖π2(b)[x−x′]‖ for t ≥ 0

with x′ = π1(b)x+m(b,π1(b)x). Let z∈M(b) be arbitrary. Then, because ofz−x′ ∈ CL(b) andπ1(b)x = π1(b)x′,

‖π2(b)[x−x′]‖ ≤ ‖π2(b)[x−z]‖+‖π2(b)[z−x′]‖= (‖π2(b)‖+L‖π1(b)‖)‖x−z‖ .

Hence‖π2(b)[x−x′]‖ ≤ (‖π2(b)‖+L‖π1(b)‖)d(x,M(b)) .

Replacingb by θ−tb, choosing aT > 0 such that

K · (‖π2(θ−tb)‖+L‖π1(θ−tb)‖)≤ eη2 t for t ≥ T

(Corollary2.8), we obtain

d(ϕ(t,θ−tb)x,M(b))≤ e−η2 td(x,M(θ−tb))≤ e−

η2 td(D,M(θ−tb))

for t ≥ T proving the pullback attractivity ofM with T independent ofD.

Corollary 2.17. If B is a metric space and ifϕ ∈C(R≥0×B×X,X) then m∈C(X1,X) withX1 := {(b,x) ∈B×X : x∈ X1(b)}.

Proof. Consider the graph transformation mappingsGT : GL ⊂ G → G on the Banach space(G,‖ · ‖G) of the continuous mappingsg∈C(X1,X),

g(b, ·) = g(b) : X1(b)→ X2(b) , b∈B

with bounded norm‖g‖G := sup

(b,x1)∈X1

‖g(b,x1)‖ ,

whereGL denotes the closed subset ofG of all functionsg satisfying the Lipschitz property

‖g(b,x1)−g(b,y1)‖ ≤ L‖x1−y1‖ for (b,x1),(b,y1) ∈ X1 .

Then the fixed-points of the graph transformation mappings are continuous functions fromX1

into X which are Lipschitz in the second argument.

16

2.3 Pullback Attractors for Nonautonomous Dynamical Systems

The following theorem shows that we can easily construct a global pullback attractor once wehave a compact, forward invariant, globally pullback absorbing set.

Theorem 2.18. [Sch92, CF94, FS96, Sch99] Suppose thatA is a compact, forward invariant,globally pullback absorbing set. ThenA,

A(b) :=⋂t≥0

ϕ(t,θ−tb)A(θ−tb) for b∈B ,

is a global pullback attractor.

Proof. SinceA is forward invariant,A(b) is the intersection of a decreasing sequence of compactsets contained inA, hence is non-void and compact, moreover

limt→∞

d(ϕ(t,θ−tb)A(θ−tb)|A(b)) = 0. (6)

Using the elementary fact that if(Dn) is a decreasing sequence of compact sets andf is continu-ous then

⋂n f (Dn) = f (

⋂nDn), the cocycle property and the monotonicity of(ϕ(t,θ−tb)A(θ−tb))

we obtain for anyT ∈ R

ϕ(T,b)A(b) =⋂t≥0

ϕ(T,b)ϕ(t,θ−tb)A(θ−tb)

=⋂t≥0

ϕ(T + t,θ−t−T(θTb))A(θ−t−T(θTb))

=⋂t≥T

ϕ(T,θ−t(θTb))A(θ−t(θTb))

=⋂t≥0

ϕ(T,θ−t(θTb))A(θ−t(θTb)) = A(θTb) ,

proving the invariance ofA.

To see thatA is globally pullback attracting choose a bounded setD ⊂ X, b∈ B and anε > 0.Due to Lemma6 there exists aT0 = T0(ε,D,b)≥ 0 such that

d(ϕ(T0,θ−T0b)A(θ−T0b)|A(b))< ε ,

and sinceA is absorbing, there exists aT1 = T1(θ−T0b,D)≥ 0 with

ϕ(t−T0,θ−(t−T0)(θ−T0b))D⊂ A(θ−T0b) if t−T0≥ T1 .

Applying ϕ(T0,θ−T0b) to this inclusion, the cocycle property implies

ϕ(t,θ−tb)D⊂ ϕ(T0,θ−T0b)A(θ−T0b) .

Using the fact that ifD1⊂D2 thend(D1|A)≤ d(D2|A), we obtain

d(ϕ(t,θ−tb)D|A(b))< ε for t ≥ T0 +T1 ,

proving thatA is globally pullback attracting.

17

The next lemma shows that we only need the existence of an absorbing setA, if we have asuitable smoothing property ofϕ. This smoothing property is expressed by the assumption thatthe setAc is compact for sufficiently largec≥ 0. A similar property is used in [Tem97] forsemiflows, and it is called there uniform compactness property.

Lemma 2.19.Suppose thatA is a globally pullback absorbing set. ThenA0,

A0(b) :=⋃s≥0

ϕ(s,θ.sb)A(θ.sb) for b∈B ,

is globally pullback absorbing and forward invariant. If there is a c≥ 0 such thatAc,

Ac(b) := cl(ϕ(c,θ−cb)A0(b)

)for b∈B ,

is compact, thenAc is a compact, forward invariant, globally pullback absorbing set.

Proof. For τ≥ 0 letAτ(b) :=

⋃s≥τ

ϕ(s,θ.sb)A(θ.sb) for b∈B .

Let b∈B, t ≥ 0. Then

ϕ(t,b)Aτ(b) =⋃s≥τ

ϕ(t +s,θ−sb)A(θ−sb)

=⋃

r≥τ+t

ϕ(r,θ−rθtb)A(θ−rθtb)

⊆⋃r≥τ

ϕ(r,θ−rθtb)A(θ−rθtb)

= Aτ(θtb) .

ThusAτ is forward invariant for eachτ≥ 0.

SinceA is globally pullback attracting, for anyb∈B and any bounded setD there is aT(b,D)with ϕ(t,θ−tb)D⊆ A(b) for t ≥ T(b,D). Sinceϕ(τ,θ−τb)A(θ−τb)⊆ Aτ(b) and

ϕ(t,θ−tθ−τb)D⊆ A(θ−τb) for t ≥ T(θ−τb,D)

we haveϕ(t,θ−tb)D⊆ Aτ(b) for t ≥ τ +T(θ−τb,D) .

ThusAτ is pullpack attracting for eachτ≥ 0. Forτ = 0 we obtain the first claim.

Now assume that there exists ac≥ 0 such thatAc is compact. Sinceϕ(c,θ−cb)A0 = Ac(b), thesetAc is the closure of the forward invariant, globally pullback attracting setAc(b) and hence itis forward invariant, globally pullback attracting, too.

18

Obviously, if A is compact, positively invariant and globally pullback attracting then we maychoosec = 0 and henceA0 = A.

In Theorem2.16we proved the existence of an inertial manifold which is exponentially attractingin the sense of forward convergence, i.e.ϕ(t,b)x is converging to its asymptotic phaseϕ(t,b)x′.Moreover we proved that the inertial manifold is globally pullback attracting if the projectorπ1

is tempered from above.

Combining the Theorems2.16 and2.18, we give a sufficient condition for the existence of aglobal pullback attractor which is contained in the inertial manifold.

Corollary 2.20. Let the assumptions of Theorem2.16be satisfied with a projectorπ1 which istempered from above. Moreover, suppose thatA is a compact, forward invariant, globally pull-back absorbing set which is tempered from above, i.e. let the function b7→ |A(b))| be temperedfrom above, where|D| := sup{‖x‖ : x∈D}.Then the global pullback attractorA,

A(b) =⋂t≥0

ϕ(t,θ−tb)A(θ−tb) b∈B ,

is contained in the inertial manifoldM.

Proof. By Theorem2.16we have the existence of the globally pullback attracting inertial mani-fold M satisfying (1). By construction there is aK > 0 with |M(b)| ≤ K for all b∈B.

Let A be a compact, forward invariant, globally pullback absorbing set which is tempered fromabove. Theorem2.18 implies the existence of the global pullback attractorA which is alsotempered from above. Choosing aT ′ > T such that|A(θ−tb)| ≤ e

η4 t for t ≥ T ′ (Corollary 2.8)

and using the invariance ofA, from (1) we get fort ≥ T ′

d(A(b)|M(b)) = d(ϕ(t,θ−tb)A(θ−tb)|M(b))≤ e−η2 t(e

η4 t +K) .

Since this expression tends to 0 fort→∞ we obtaind(A(b)|M(b)) = 0 and conclude thatA(b)⊂M(b) for all b∈B.

3 Nonautonomous Evolution Equations

3.1 Two-Parameter Semi-Flow

Let (X,‖ ·‖X) be a Banach space. The solutions of a nonautonomous evolution equation will notgenerate a semi-flow but a two-parameter semi-flow.

Definition 3.1 (Two-parameter Semi-Flow).A two-parameter semi-flow µonX is a continuousmapping

{(t,s,x) ∈ R×R×X : t ≥ s} 3 (t,s,x) 7→ µ(t,s,x) ∈ X

19

which satisfies

(i) µ(s,s, ·) = idX for s∈ R;

(ii) the two-parameter semi-flow property fort ≥ τ≥ s, x∈ X, i.e.

µ(t,τ,µ(τ,s,x)) = µ(t,s,x) .

The next lemma explains how a two-parameter semi-flow defines an NDS.

Lemma 3.2 (Two-parameter Semi-Flow defines NDS).Suppose that µ is a two-parametersemi-flow. Thenϕ : R≥0×B×X→ X,

ϕ(t,b)x = µ(t +b,b,x) (7)

is an NDS with baseB = R and driving systemθ : R×B→B,

θ(t)b = t +b.

Moreover, for t≥ s and x∈ X the relation µ(t,s,x) = ϕ(t−s,s)x holds.

Proof. θ is a dynamical system. We have

ϕ(0,b) = µ(b,b, ·) = idX .

We use the two-parameter semi-flow property ofµ to obtain fort,s≥ 0, b∈B

ϕ(t +s,b) = µ(t +s+b,b, ·)= µ(t +s+b,s+b,µ(s+b,b, ·))= ϕ(t,θsb)◦ϕ(s,b) ,

proving the cocycle property ofϕ. The continuity ofµ implies the continuity of(t,x) 7→ ϕ(t,b)x.Now substitutet by t−s andb by s in (7) to see thatµ(t,s,x) = ϕ(t−s,s)x.

Theorem 3.3 (Inertial Manifold for Two-parameter Semi-Flow). Suppose that µ is a two-parameter semi-flow onX and let(πi(τ))τ∈R ⊂ L(X), i = 1,2, be two families of complementaryprojectorsπ1(τ) andπ2(τ). Let µ satisfy the

• cone invariance property, i.e. there are L> 0 and T0≥ 0 such that forτ ∈ R and x,y∈ X,

x−y∈ CL(τ) := {ξ : ‖π2(τ)ξ‖X ≤ L‖π1(τ)ξ‖X}

impliesµ(t,τ,x)−µ(t,τ,y) ∈ CL(t) for t ≥ τ +T0 ;

20

• squeezing property, i.e. there exist positive constants K1, K2 andη such that for everyτ ∈ R,x,y∈ X and T> 0 the identity

π1(τ +T)µ(τ +T,τ,x) = π1(τ +T)µ(τ +T,τ,y)

implies for all x′ ∈ X with π1(τ)x′ = π1(τ)x and x′−y∈ CL(τ) the estimates

‖πi(t) [µ(t,τ,x)−µ(t,τ,y)]‖X ≤ Kie−η(t−τ)‖π2(τ)

[x−x′

]‖X , i = 1,2,

for t ∈ [τ,τ +T];

• boundedness property, i.e. for all t,τ ∈ R with t≥ τ and all M1≥ 0 there exists a M2≥ 0 suchthat for x∈ X with ‖π2(τ)x‖X ≤M1 the estimate

‖π2(t)µ(t,τ,x)‖X ≤M2

holds.

• coercivity property, i.e. for all t,τ ∈ R with t ≥ τ and all M3 ≥ 0 there exists a M4 ≥ 0 suchthat for x∈ X with ‖π1(τ)x‖X ≥M4 the estimate

‖π1(t)µ(t,τ,x)‖X ≥M3

holds.

Then there exists an inertial manifoldM = (M(τ))τ∈R of µ with the following properties:

(i) M(τ) is a graph inπ1(τ)X⊕π2(τ)X,

M(τ) = {x1 +m(τ,x1) : x1 ∈ π2(τ)X}

with a mapping m(τ, ·) = m(τ) : π1(τ)X→ π2(τ)X which is globally Lipschitz continuous

‖m(τ,x1)−m(τ,y1)‖X ≤ L‖x1−y1‖X .

(ii) M is exponentially attracting, more precisely

‖πi(t)[µ(t,τ,x)−µ(t,τ,x′)]‖X ≤ Kie−η(t−τ)‖π2(τ)x−m(τ,π1(τ)x)‖X

for t ≥ τ, i = 1,2, with an asymptotic phase x′ = x′(τ,x)∈M(τ) of x and the constants K1,K2> 0from the squeezing property which are independent ofτ and x.

Proof. By Lemma3.2, the two-parameter semi-flowµ defines an NDSϕ with baseB = R anddriving systemθ : R×B→ B with θ(t)τ = t + τ, τ = b∈ B. Now Theorem3.3 follows fromTheorem2.16.

21

3.2 Nonautonomous Evolution Equations

Let X ⊆ Y be Banach spaces equipped with norms‖ · ‖X, ‖ · ‖Y, and let(A(t))t∈R be a familyof densely defined linear operatorsA(t) on Y with domainD(A(t)) in Y. We consider a nonau-tonomous evolution equation

x+A(t)x = f (t,x) (8)

which satisfies the following assumptions:

(A1) Linearity A(t):

• Existence of evolution operator of the linear system: Under suitable additional assumptions onX, Y, A and f (see for example [Hen81, DKM92, Lun95]), we may define the evolution operatorΦ : {(t,s) ∈ R2 : t ≥ s} → L(Y,Y) of the linear equation

x+A(t)x = 0 (9)

in Y as the solution of

ddt

Φ(t,s)+A(t)Φ(t,s) = 0 for t > s, s∈ R

andΦ(τ,τ) = idY for τ ∈ R .

• There are constantsk0, . . . ,k4 ≥ 1, β2 > β1, α ∈ [0,1[, a monotonously decreasing functionψ ∈C(R>0,R>0) with ψ(t)≤ k0t−α, and a familyπ1 = (π1(t))t∈R of linear, invariant projectorsπ1(t) : Y→ Y, i.e.

π1(t)Φ(t,s) = Φ(t,s)π1(s) for t ≥ s,

such thatΦ(t,s)π1(s) can be extended to a linear, bounded operator fort ∈ R with

ddt

Φ(t,s)π1(s)+A(t)Φ(t,s)π1(s) = 0 for t,s∈ R

and‖Φ(t,s)π1(s)‖L(X,X) ≤ k1e−β1(t−s) for t ≤ s,

‖Φ(t,s)π2(s)‖L(X,X) ≤ k2e−β2(t−s) for t ≥ s,

‖Φ(t,s)π1(s)‖L(Y,X) ≤ k3e−β1(t−s) for t ≤ s,

‖Φ(t,s)π2(s)‖L(Y,X) ≤ k4ψ(t−s)e−β2(t−s) for t > s

(10)

with π2, π2(t) = idY−π1(t), as the complementary projector toπ1 in Y.

(A2) Nonlinearity f(t,x): The nonlinear functionf ∈ C(R×X,Y) is assumed to be globallybounded and to satisfy the Lipschitz inequality

‖πi(t)[ f (t,x)− f (t,y)]‖Y ≤ γi(‖π1(t)[x−y]‖X,‖π2(t)[x−y]‖X) (11)

22

for all t ∈ R, x,y∈ X, whereγi are suitable norms onR2.

(A3) Existence of mild solutions:We have the existence and uniqueness of the mild solutions

µ(·,τ,ξ) ∈C([τ,∞[,X)

of (8) with initial condition x(τ) = ξ ∈ X, i.e. let µ be the continuous solution of the integralequation

x(t) = Φ(t,τ)ξ +∫ t

τΦ(t, r) f (r,x(r))dr for t ≥ τ .

These were the assumptions.

Remark3.4. Conditions like our assumptions can be found in the literature and they are stan-dard for ordinary differential equations and for time-independent evolution equations in the non-selfadjoint case, see for example [Tem97]. For concrete examples of the realization of theseassumptions we refer to Sec.4, where we will apply our following Theorem3.10on the exis-tence of inertial manifolds in these special situations.

3.3 Comparison Theorems

In order to apply Theorem3.3, we have to show the cone invariance property and the squeezingproperty for the two-parameter semi-flowµ with respect to the projectorπ1.

For fixedr1, r2≥ 0 andT ≥ 0, we define

(Λ1w)(t) := k3

∫ T

te−β1(t−r)

1 γ1(w(r))dr ,

(Λ2w)(t) := k4

∫ t

0ψ(t− r)e−β2(t−r)γ2(w(r))dr +k2e−β2tLw1(0)

andq(t) :=

(k1e−β1(t−T)r1,k2e−β2tr2

)for t ∈ [0,T] andw ∈ C([0,T],R2). Thenq ∈ C([0,T],R2

≥0). Because ofψ(t) ≤ k0t−α withα ∈ [0,1[, Λ is an at most weakly singular integral operator fromC([0,T],R2) intoC([0,T],R2).Moreover,Λ is completely continuous.

Lemma 3.5. Assume there are L> 0 and T0≥ 0 such that for each T≥ T0 the inequality

v2(T)≤ Lr1 (12)

holds for each solution v∈C([0,T],R2≥0) of

vi(t)≤ (Λv)i (t)+qi(t) for i = 1,2, t ∈ [0,T] (13)

with r1 ≥ 0, r2 = 0. Then µ possesses the cone invariance property with respect toπ with theparameters L> 0 and T0≥ 0.

23

Proof. The cone invariance property is shown if

‖π2(t)[µ(t,τ,x)−µ(t,τ,y)]‖X ≤ L‖π1(t)[µ(t,τ,x)−µ(t,τ,y)]‖X (14)

for eachτ ∈ R, t ≥ τ +T0 andx,y∈ X with

‖π2(τ)[x−y]‖X ≤ L‖π1(τ)[x−y]‖X . (15)

Let τ ∈ R, T > 0 andx,y∈ X with (15) be fixed and let

λ(t) := µ(t,τ,x)−µ(t,τ,y) , f∆(t) := f (t,µ(t,τ,x))− f (t,µ(t,τ,y))

on [τ,τ +T]. Then we have

π1(t)λ(t) = Φ(t,τ +T)π1(τ +T)λ(τ +T)+∫ t

τ+TΦ(t, r)π1(r) f∆(r)dr

and

π2(t)λ(t) = Φ(t,τ)π2(τ)[x−y]+∫ t

τΦ(t, r)π2(r) f∆(r)dr

on [τ,τ +T]. Let v : [0,T]→ R2≥0 be defined by

v(t) = (‖π1(τ + t)λ(τ + t)‖X,‖π2(τ + t)λ(τ + t)]‖X) . (16)

Then

v1(t)≤ ‖Φ(τ + t,τ +T)π1(τ +T)λ(τ +T)‖X

+∫ T

t‖Φ(τ + t,τ + r)π1(τ + r)∆ f (τ + r)‖X dr

≤ ‖Φ(τ + t,τ +T)π1(τ +T)‖L(X,X)‖π1(τ +T)λ(τ +T)‖X

+∫ T

t‖Φ(τ + t,τ + r)π1(τ + r)‖L(Y,X)‖π1(τ + r)∆ f (τ + r)‖Y dr

and

v2(t)≤ ‖Φ(τ + t,τ)π2(τ)λ(τ)‖X

+∫ t

0‖Φ(τ + t,τ + r)π2(τ + r)∆ f (τ + r)‖X dr

≤ ‖Φ(τ + t,τ)π2(τ)‖L(X,X)‖π2(τ)λ(τ)‖X

+∫ t

0‖Φ(τ + t,τ + r)π2(τ + r)‖L(Y,X)‖π2(τ + r)∆ f (τ + r)‖Y dr .

The exponential dichotomy and Lipschitz assumptions (10), (11) and inequality (15) imply thatthe functionv defined by (16) satisfies (13) with

r1 := ‖π1(τ)[µ(τ +T,τ,x)−µ(τ +T,τ,y)]‖X , r2 := 0.

By the assumptions we have (15), i.e. (14) holds.

24

Lemma 3.6. Assume there are positive numbers L,η, K1, K2 such that

vi(t)≤ Kie−ηtr2 for t ∈ [0,T] (17)

holds for each T> 0 and each solution v∈C([0,T],R2≥0) of (13) with r1 = 0, r2 ≥ 0. Then µ

possesses the squeezing property with respect toπ with the parameters L,η, K1, K2.

Proof. The squeezing property is shown if we findη > 0 and constantsK1,K2 > 0 such that forall τ ∈ R, x,y∈ X andT > 0 the identity

π1(τ +T)µ(τ +T,τ,x) = π1(τ +T)µ(τ +T,τ,y) (18)

implies for allx′ ∈ X with

π1(τ)x′ = π1(τ)x and ‖π2(τ)[x′−y]‖X ≤ L‖π1(τ)[x′−y]‖X (19)

the estimates

‖πi(t)[µ(t,τ,x)−µ(t,τ,y)]‖X ≤ Kie−η(t−τ)‖π2(τ)[x−x′]‖X (20)

for i = 1,2 andt ∈ [τ,τ +T].

As in the proof of Lemma3.5 let τ ∈ R, T > 0 andx,y,x′ ∈ X with (18), (19) be fixed and let

λ(t) := µ(t,τ,x)−µ(t,τ,y) , f∆(t) := f (t,µ(t,τ,x))− f (t,µ(t,τ,y))

on [τ,τ +T]. Then we have

π1(t)λ(t) = Φ(t,τ +T)π1(τ +T)λ(τ +T)+∫ t

τ+TΦ(t, r)π1(r) f∆(r)dr

=∫ t

τ+TΦ(t, r)π1(r) f∆(r)dr

and

π2(t)λ(t) = Φ(t,τ)π2(τ)[x′−y]+ Φ(t,τ)π2(τ)[x−x′]

+∫ t

τΦ(t, r)π2(r) f∆(r)dr

on [τ,τ + T]. Similar to the proof of Lemma3.5, the exponential dichotomy and Lipschitz as-sumptions (10), (11) and inequality (19) imply that the functionv : [0,T]→R2

≥0 defined by (16)satisfies (13) with

r1 := 0, r2 := ‖π2(τ)[x′−x]‖X .

By assumption we have (17), i.e. (20) holds.

25

To estimate the solutionsv of (13), we make an excurse to the theory of monotone iterations inordered Banach spaces, see for example [KLS89].

Let B be a Banach space and letC be an order cone inB. The order coneC induces a semi-order≤C in B by

u≤C w :⇐⇒ w−u∈ C .

The norm inB is calledsemi-monotoneif there is a constantc with ‖x‖B ≤ c‖y‖B for eachx,y∈B with 0≤C x≤C y. The coneC is called normal if the norm in is semi-monotone, andC

is called solid ifC contains an open ball with positive radius.

Note thatC([0,T],RN≥0) is a normal, solid cone inC([0,T],RN).

In a Banach spaceB with normal and solid coneC, we study the fixed-point problem

u = Pu+ p (21)

with p∈B andP : B→B. We assume thatP is completely continuous, increasing,

Pu≤C Pv if u≤C v,

subadditive,P(u+v)≤C Pu+Pv, u,v∈ C ,

and homogeneous with respect to nonnegative factors,

P(λu) = λPu λ ∈ R≥0 , u∈ C .

Definition 3.7. A functionw∈B is calledupper (lower) solutionof (21) if Pw+ p≤C w (w≤C

Pw+ p).

Our goal is to ensure the existence of a unique solutionw ∈ C of (21) and to estimate lowersolutionsv∈ C of (21) by solutions or upper solutions of (21).

For x,y∈B with x≤C y, we denote by

[x,y]C := {z∈B : x≤C z≤C≤ y}

the order interval given byx andy.

Lemma 3.8. Let x0 ∈ C andx0 ∈ C be a lower and an upper solution of(21) with x0 ≤C≤ x0.Then the sequences(xn)n∈N and(xn)n∈N with xn+1 = Pxn + p andxn+1 = Pxn + p converges tosolutions x∗ andx∗ of (21) and

x0≤C x∗ ≤C x∗ ≤C x0 . (22)

26

Proof. We follow the proof of Theorem 3.1 in [EL75].

By the normality of the cone, there is a constantc> 0 such that‖w1‖B≤ c‖w2‖B for all w1, w2∈C satisfyingw1 ≤C w2. Hence, the sets[x0,x0]C ⊆ [0,x0]C are norm bounded byc‖x0‖B andclosed. Because of the complete continuity ofP, there is a convergent subsequence

(xnk

)k∈N of

(xn)n∈N in [x0,x0]C with limit x∗ ∈ [x0,x0]C. Suppose(xn)n∈N is not convergent tox∗. Then thereareδ> 0 and a subsequence

(xmk

)k∈N of (xn)n∈N satisfying

‖xmk−x∗‖B > δ , xmk

≤C x∗ , xnk≤C xmk

for k∈ N .

By the normality of the cone, we have

‖x∗−xmk‖B ≤C c‖x∗−xnk

‖B→ 0

for k→ ∞ and a contradiction is reached. Thus,(xn)n∈N is convergent tox∗.

Analogously, one can show the converges of(xn)n∈N.

By construction we have

x0≤C x1≤C x2≤C · · · ≤C x∗ ≤C x∗ ≤C · · · ≤C x2≤C x1≤C x0 .

The inequalitiesxn ≤C xn+1 = Pxn + p≤C x∗, the convergencexn→ x∗ and the continuity ofPimply

x∗ ≤C Px∗+ p≤C x∗ ,

i.e.x∗ is a solution of (21).

Similarly follows thatx∗ is a solution of (21), too. By construction we have (22).

Lemma 3.9. Assume that there are y∈ intC andδ ∈ [0,1[ with

Py≤C δy.

Then there is a unique solution x∗ of (21) in C and

x≤C x∗ ≤C x (23)

holds for each lower solution x∈ C and each upper solutionx∈ C of (21).

Proof. First we note that by the normality and solidity ofC, for eachx∈ C there is‖x‖y definedby

‖x‖y := inf{µ≥ 0 : −µy≤C x≤C µy} .

Now we show that 0 is the unique solution of

x = Px (24)

27

in C. SinceP is subadditive, we haveP0 = P(0+ 0) ≤C P0+ P0 and hence 0≤C P0. From0≤C y follows 0≤ P0≤C Py≤C δy, i.e. 0≤C P0≤C δy. Thus 0≤C P0≤C δny for all n∈ N.Hence 0= P0. Let x ∈ C be a solution of (24). Then 0≤C x≤C µ(x)y. By the monotony andhomogeneity ofP we have 0≤C x≤C ‖x‖yδny for all n∈ N and hencex = 0. Therefore, 0 is theunique solution of (24) in C.

By the subadditivity ofP and because ofp∈ C, 0 is a lower solution of (21). Moreoverλy with

λ≥ ‖q‖y1−δ is an upper solution of (21). By Lemma3.8, the existence of at least one solutionx1 ∈ C

of (21) follows.

Assume there is another solutionx2 ∈ C of (21). By Lemma3.8, we find a solutionx3 ∈ C of

(21) with 0≤C xi ≤C≤ x3 ≤C λy for i = 1,2 with λ ≥max{‖x1‖y,‖x2‖y,‖q‖y1−δ}. Hence, without

loss of generality we may assume that 0≤C x1≤C≤ x2 andx1 6= x2.

Let w := x2−x1. Then 0≤C w, w 6= 0, and, by the subadditivity ofP,

w = Px2−Px1 = P(x1 +w)−Px1≤C Px1 +Pw−Px1 = Pw.

Thus,w is a lower solution of (24). Sinceµ(w)y is an upper solution of (24) andw≤C ‖w‖yy,Lemma3.8 with p = 0 implies the existence of a solutionv∈ [w,y]C of (24). As above shown,v = 0 is the unique solution of (24) in C. Hencew = 0 in contrast to the assumptionx1 6= x2.Thus (21) possesses a unique solutionx∗ ∈ C.

Now let x∈ C, x∈ C be arbitrary lower and upper solutions of (21). By Lemma3.8 and by the

uniqueness of the solutionx∗ of (21), we havex≤C x∗ ≤C λy with λ ≥max{‖x‖y,‖x∗‖y,‖q‖y1−δ}.

On the other hand, with the lower solution 0 of (21), we have 0≤C x∗ ≤C x. Thus (23) holds.

In order to apply Lemma3.9to our situation, we chooseB =C([0,T],R2) andC =C([0,T],R2≥0).

ThenC is a normal. The operatorP = Λ is increasing and completely continuous, andp = q be-longs toC. So we only have to find a functionw∗ in the interior ofC with

Λw∗ ≤C εw∗ with someε ∈ [0,1[ . (25)

Further we can estimate the solutionsv of (13) by solutions ¯w∈ C of

Λw+q≤C w . (26)

3.4 Inertial Manifold Theorem for Nonautonomous Evolution Equations

Now we are in a position to prove the following theorem.

Theorem 3.10 (Inertial Manifolds for Nonautonmous Evolution Equations).Under the gen-eral assumptions in Sec.3.2, let t∗ ≥ 0 be fixed with

ψ∗ := limt→t∗

ψ(t)< ∞ , ψ(t)> ψ∗ for t < t∗ . (27)

28

Further let

k5≥ δ∫ t∗

0ψ(r)e−δr dr + ψ∗ lim

t→t∗e−δt for all δ ∈ ]0,β2−β1[ . (28)

Assume that there are positive numbersρ1 < ρ2 with

G(ρ1) = G(ρ2) = 0, G∣∣[ρ1,ρ2] 6= 0 (29)

andk1k2ρ1 < k−1

5 ψ∗ρ2 (30)

where G: R>0→ R is defined by

G(ρ) := β2−β1−k3γ1(1,ρ)−k4k5ρ−1γ2(1,ρ) .

Then there are positive numbersη1 < η2 with

ηi = β1 +k3γ1(1,ρi) = β2−k4k5ρ−1i γ2(1,ρi) ,

and the claim of Theorem3.2holds for the two-parameter semi-flow µ generated by(8) with

η := η2 , L ∈ ]k1ρ1,k−12 k−1

5 ψ∗ρ2[ , (31)

and

K1 :=k2k5

ρ2ψ∗−k2k5L, K2 := ρ2K1 . (32)

Remark3.11. Because of limρ→0G(ρ) = limρ→∞ G(ρ) = −∞, the existence ofρ∗ > 0 withG(ρ∗) > 0 implies the existence of positive numbersρ1 < ρ2 with (29). SinceG(ρ1) = 0 andρ1 < ρ∗ imply

ρ1 <k4k5ρ−1

∗ γ2(1,ρ∗)β2−β1−k3γ1(1,ρ∗)

,

the inequality (30) holds if

β2−β1 > k3γ1(1,ρ∗)+k1k2k4k2

5

ψ∗ρ∗γ2(1,ρ∗) . (33)

Since (33) impliesG(ρ∗)> 0, the conditions (29), (30) can be replaced by (33) for someρ∗ > 0.

Proof. (of Theorem3.10) We show that the two-parameter semiflowµ generated by (8) satisfiesthe assumptions of Theorem3.2. We split the proof into five parts: First we show that (29)implies the existence of solutions of (25) and (26). Then we verify the four required propertiesfor the two-parameter semiflowµ.

Step 1: Determining of Solutions of (25) and (26)

29

In order to find a solutionw∗ of (25), first we look forw∈ C in the form

w(t) = e−ηt(1,ρ) (34)

with ρ> 0 and satisfyingw1(t)≥ (Λw)1(t)+c1e−β1(t−T) (35)

andw2(t)≥ (Λw)2(t)+(c2ρ−k2L)e−β2t (36)

for t ∈ [0,T] with suitable positivec1 andc2.

If we assumeη> β1 andη≥ β1 +k3γ1(1,ρ) (37)

then, because of

eηt (Λw)1(t) = k3γ1(1,ρ)∫ T

te(η−β1)(t−r) dr

=k3γ1(1,ρ)

η−β1

(1−e(η−β1)(t−T)

),

we may choose

c1 = c1(ρ,η) :=k3γ1(1,ρ)

η−β1e−ηT (38)

in order to satisfy (35). Remains to satisfy (36).

Inserting (34) in (36) and dividing byρe−ηt , we have to satisfy

1≥ H(t,ρ,η) (39)

for t ∈ [0,T] whereH : [0,∞[×R×R→ R is defined by

H(t,ρ,η) :=γ2(1,ρ)

ρk4

∫ t

0ψ(r)e−(β2−η)r dr +c2e−(β2−η)t .

We choose

c2 = c2(ρ,η) :=γ2(1,ρ)

(β2−η)ρk4ψ∗ .

Because of

D1H(t,ρ,η) =(−(β2−η)c2 +

γ2(1,ρ)ρ

k4ψ(t))

e−(β2−η)t

and because of the monotonicity ofψ, the functionH(·,ρ,η) is maximized att∗. Hence we have

H(t,ρ,η)≤ H(t∗,ρ,η)

=γ2(1,ρ)

ρk4

(∫ t∗

0ψ(r)e−(β2−η)rdr +(β2−η)−1ψ∗e−(β2−η)t∗

)

30

for all t ≥ 0. Because of (28), inequality (39) is satisfied if

γ2(1,ρ)ρ

k4k5≤ β2−η . (40)

Combining (37) with (40), we find

β1 +k3γ1(1,ρ)≤ η≤ β2−k4k5ρ−1γ2(1,ρ) (41)

as a sufficient condition for (35) and (36).

By assumption there are positive numbersρ1 < ρ2 with (29) and (30). Let

ηi := β1 +k3γ1(1,ρi) = β2−k4k5ρ−1γ2(1,ρi) .

Then(η1,ρ1) and(η2,ρ2) solve (41).

Moreover,η2 > η1. To show this, we note thatη2 ≥ η1 by the monotonicity ofγ1. Assumingη1 = η2 we find γ1(1,ρ1) = γ1(1,ρ2) andγ2(ρ−1

1 ,1) = γ2(ρ−12 ,1). By the convexity of theγ1-

andγ2-balls we hadγ1(1,ρ) = γ1(1,ρ1) andγ2(ρ−1,1) = γ2(ρ−11 ,1) for ρ ∈ [ρ1,ρ2]. This would

imply the constance ofG on [ρ1,ρ2] in contradiction to (29).

Because of (30) we can choose

L ∈ ]k1ρ1,k−12 k−1

5 ψ∗ρ2[ . (42)

Now we definewi ∈ C, i = 1,2, by

wi(t) := e−ηit(1,ρi) . (43)

Then

w1i (t)≥ (Λwi)

1(t)+k3γ1(1,ρ)

η−β1e−ηTe−β1(t−T)

andw2

i (t)≥ (Λwi)2(t)+

(ρik−15 ψ∗−k2L

)e−β2t

on [0,T] because of

c1(ρi ,ηi) =k3γ1(1,ρi)

ηi−β1= 1, c2(ρi ,ηi) =

γ2(1,ρi)k4ψ∗(β2−ηi)ρi

= k−15 ψ∗ .

Because of (42) we haveρ2k−1

5 ψ∗ > k2L (44)

and inequality (25) holds forw∗ := w2.

31

Let nowC1 ∈ [0,1], C2 > 0 satisfy

C2(C1e−η1T +(1−C1)e−η2T)≥ k1r1 ,

C2

(C1ρ1k−1

5 ψ∗+(1−C1)ρ2k−15 ψ∗−k2L

)≥ k2r2 . (45)

Thenw := C2(C1w1 +(1−C1)w2)

solvesΛw+q≤C w ,

and Lemma3.9 impliesv≤C w

for each solutionv∈ C of (13).

Step 2: Verification of the Cone Invariance Property

Because of (44) we can fixρ ∈ ]max{ρ1,k2k5ψ−1

∗ L},ρ2[ .

Let r2 = 0 andr1≥ 0. Then (45) is satisfied with

C1 :=ρ2− ρρ2−ρ1

, C2 :=k1r1

C1e−η1T +(1−C1)e−η2T .

Thus we findv2(t)≤ w2(t) = C2

(C1w2

1(t)+(1−C1)w22(t))

= L(ρ, t)r1

for t ∈ [0,T] with

L(ρ, t) = k1(ρ2− ρ)ρ1e−η1t +(ρ−ρ1)ρ2e−η2t

(ρ2− ρ)e−η1T +(ρ−ρ1)e−η2T .

Especially we havev2(T)≤ L(ρ,T)r1 .

The inequalitiesη2 > η1 > β1 imply

L(ρ,T)→ k1ρ1 asT→ ∞ .

Hence there areT0≥ 0 andL≥ 0 with (12), if the additional inequality

k1ρ1 < L

holds, which trivially follows from (42).

By Lemma3.5, the cone invariance property ofµ as required in Theorem3.3 is verified.

32

Step 3: Verification of the Squeezing Property

Now let r1 = 0 andr2≥ 0. Then we may choose

C1 := 0, C2 :=k2k5r2

ρ2ψ∗−k2k5L

in order to satisfy (45). Thus we find

v(t)≤ k2r2

ρ2ψ∗−k2k5Le−η2t(1,ρ2) for t ∈ [0,T] .

Hence (17) holds with

η := η2 , K1 :=k2k5

ρ2ψ∗−k2k5L, K2 := ρ2K1

andL satisfying (42). By Lemma3.6, the squeezing property ofµ as required in Theorem3.3 isverified.

Step 4: Verification of the Boundedness Property

By the boundedness off there is a numberF ≥ 0 with

‖ f (x)‖Y ≤ F for x∈ X .

Thus, forτ ∈ R, t ≥ τ, x∈ X,

π2(t)µ(t,τ,x) = Φ(t,τ)π2(τ)x+∫ t

τΦ(t, r)π2(r) f (r,µ(r,τ,x))dr

and, by the exponential dichotomy conditions (10),

‖π2(t)µ(t,τ,x)‖X ≤ ‖Φ(t,τ)π2(τ)‖L(X,X)‖π2(τ)x‖X

+∫ t

τ‖Φ(t, r)π2(r)‖L(Y,X)‖ f (r,µ(r,τ,x))‖X dr

≤ k2e−β2(t−τ)‖π2(τ)x‖X +Fk4

∫ t

τψ(t− r)e−β2(t−r) dr

= k2e−β2(t−τ)‖π2(τ)x‖X +Fk4

∫ t−τ

0ψ(r)e−β2(r) dr

≤ k2e−β2(t−τ)‖π2(τ)x‖X +Fk4

∫ ∞

0ψ(r)e−β2(r) dr .

Thus, for anyt,τ with t ≥ τ and anyM1 ≥ 0 there is anM2 ≥ 0 such that forx ∈ X with‖π2(τ)x‖X ≤ M1 we have‖π2(t)µ(t,τ,x)‖X ≤ M2, i.e. the two-parameter flow possesses theboundedness property ofµ as required in Theorem3.2.

33

Step 5: Verification of the Coercivity Property

For τ ∈ R, t ∈ [τ,τ +T], x∈ X, we have

π1(t)µ(t,τ,x) = Φ(t,τ +T)π1(τ +T)µ(τ +T,τ,x)

+∫ t

τ+TΦ(t, r)π1(r) f (r,µ(r,τ,x))dr

and hence

π1(τ)x = Φ(τ,τ +T)π1(τ +T)µ(τ +T,τ,x)

+∫ τ

τ+TΦ(τ, r)π1(r) f (r,µ(r,τ,x))dr .

The exponential dichotomy conditions (10) imply

‖π1(τ)x‖X ≤ ‖Φ(τ,τ +T)π1(τ +T)‖L(X,X)‖π1(τ +T)µ(τ +T,τ,x)‖X

+F∫ τ+T

τ‖Φ(τ, r)π1(r)‖L(Y,X) dr

≤ k1eβ1T‖π1(τ +T)µ(τ +T,τ,x)‖X +Fk3

∫ τ+T

τe−β1(τ−r) dr

= k1eβ1T‖π1(τ +T)µ(τ +T,τ,x)‖X +Fk3

β1(eβ1T −1) .

Hence

‖π1(τ +T)µ(τ +T,τ,x)‖X ≥1k1‖π1(τ)x‖X−

Fk3

β1k1(1−e−β1T)

for T ≥ 0, τ ∈ R, x ∈ X which shows the coercivity property of the two-parameter flowµ asrequired in Theorem3.2.

Thus all assumption of Theorem3.2 are verified and Theorem3.2 implies the existence of aninertial manifold with the stated properties.

3.5 Pullback Attractors for Nonautonomous Evolution Equations

In addition to the assumptions of Subsection3.2we suppose:

• There is a Banach spaceZ which is compactly embedded inX.

• There arek7, γ ∈ ]0,1[ andβ0 > 0 such that the evolution operatorΦ of (8) mapsY into Z andsatisfies

‖Φ(t,s)‖L(X,X) ≤ k7e−β0(t−s) ,

‖Φ(t,s)‖L(Y,X) ≤ k7(t−s)αe−β0(t−s) ,

‖Φ(t,s)‖L(Y,Z) ≤ k7(t−s)γe−β0(t−s) for t > s.

34

• There is an 0≥ 0 with

‖ f (t,x)‖Y ≤ `0 for (t,x) ∈ R×X .

Theorem 3.12.Under the above assumptions there is a global pullback attractorA = (A(τ))τ∈Rfor (8). If π1 is tempered from above thenA is contained in the inertial manifoldM.

Remark3.13. The assumption of Theorem3.12on the linear partA(t) of (8) are easily to sat-isfy if A is a time-independent, selfadjoint positive operator on the Hilbert spaceY. In depen-dence on the nonlinearityf , the Hilbert spaceX can be chosen as the domainD(Aα) of a powerAα of A with α ∈ [0,1[. Further, we may chooseZ = D(Aγ) with someγ ∈ ]α,1[. SinceA istime-independent, the projectorπ1 from Y onto the subspace spanned by the firstN eigenvec-tors e1, . . . ,eN of A belonging to theN smallest eigenvaluesλ1 ≤ ·· · ≤ λN of A is also time-independent and hence trivially tempered from above.

Proof. (of Theorem3.12) Let µ denote the two-parameter semiflow generated by (8). Thusϕ(t,τ)x = µ(t + τ,τ,x) and

ϕ(t,τ)x = Φ(t + τ,τ)ξ +∫ t

0Φ(t + τ,s+ τ) f (τ +s,ϕ(s,τ)x)ds.

By Lemma2.19it is sufficient to construct a globally pullback absorbing setA such thatA1,

A1(τ) := cl

(ϕ(1,τ−1)

⋃s≥0

ϕ(s,τ−s)A(τ−s)

)for τ ∈ R ,

is compact.

For t ≥ 0, τ ∈ R andx∈ X, we estimate

‖ϕ(t,τ)x‖X ≤ ‖Φ(t + τ,τ)‖L(X,X)‖x‖X

+∫ t

0‖Φ(t + τ,s+ τ)‖L(Y,X)‖ f (τ +s,ϕ(s,τ)x)‖Y ds

≤ k7e−β0t‖x‖X + `0k7

∫ t

0t−αe−β0t dr

≤ k7e−β0t‖x‖X + `0k7k8

withk8 :=

∫ ∞

0t−αe−β0t dr .

Let r0 > `0k7k8 andA(τ) := {x∈ X : ‖x‖X ≤ r0} for τ ∈ R .

ThenA is globally pullback attracting,

ϕ(t,τ− t)D⊆ A(τ)) for all t ≥ T(|D|) ,

35

with

|D|Y := supx∈D‖x‖Y , T(r) := β−1

0 ln

(k7r

r0− `0k7k8

).

Let τ ∈ R. We show thatA1(τ) are compact subsets ofX. First we note that, fors≥ 0, ϕ(s,τ−s)A(τ− s) is bounded inX by r1 := k7r0 + `0k7k8. ThusA0(τ) :=

⋃s≥0ϕ(s,τ− s)A(τ− s) is

bounded inX by r1. Let x∈ X with ‖x‖X ≤ r1. Then

‖ϕ(1,τ−1)x)‖Z≤ ‖Φ(τ,τ−1)‖L(Y,Z)‖x‖Y

+∫ 1

0‖Φ(τ,s+ τ−1)‖L(Y,Z)‖ f (s+ τ−1,ϕ(s,τ−1)x)‖Y ds

≤ k7e−β0r1 + `0k7

∫ 1

0(1−s)−γe−β0(1−s) ds

= k7e−β0r1 + `0k7

∫ 1

0se−β0sds=: r2 .

Henceϕ(1,τ−1)A0(τ) is bounded inZ by r2. SinceZ is compactly embedded inX, the closureA1(τ) of ϕ(1,τ−1)A0(τ) in X is a compact subset ofX for all τ ∈ R.

Remains to show thatA1 is tempered from above. Sinceϕ(1,τ−1)A0(τ)⊆ A0(τ), the setA1(τ)is bounded inX by r1 for all τ ∈ R. HenceA1 is trivially tempered from above.

4 Conclusion

Exponential dichotomy conditions of the form (10) are used, for example, in [Hen81], [Tem97],[BdMCR98], [LL99], [CS01]. Therek3 = βα

1k1, k4 = βα2 with someα ∈ [0,1[ depending on the

spacesX andY, andψ(t) = β−α2 max{t−α,1}, ψ(t) = β−α

2 t−α +1, orψ(t) = max{ααβ−α2 t−α,1}

where 00 := 1. If A is a time-independent sectorial operator, then usuallyX is the domainD((A+a)α) of the power(A+a)α of A+a with someα ∈ [0,1[ and somea∈R. If X = Y then we maychooseα = 0 andψ = 1.

In the special case thatA is a time-independent, selfadjoint positive linear operator with compactresolvent and dense domainD(A) on the Hilbert spaceY, usually one usesX = D(Aα) withsomeα ∈ [0,1[. Let π1 be the orthogonal projector fromY onto the linear subspace spanned bythe N eigenvectors ofA corresponding to the firstN eigenvaluesλ1 ≤ ·· · ≤ λN (counted withtheir multiplicity). Then we may chooseβ1 = λN, β2 = λN+1, k1 = k2 = 1, k3 = βα

1 , k4 = βα2 ,

ψ(t) := max{ααβ−α2 t−α,1}, see for example [FST88, Lemma 3.1].

In [LL99] (and withX = Y), a Lipschitz inequality of the form

‖πi(t)[ f (t,x)− f (t,y)]‖X ≤ `i max{‖π1(t)[x−y]‖X,‖π2(t)[x−y]‖X}

36

is utilized. This special form of a Lipschitz inequality is contained in our Lipschitz assumptionwith γi(w) = `i |w|∞ and| · |∞ as the maximum norm inR2. The standard Lipschitz inequality

‖ f (t,x)− f (t,y)‖Y ≤ `‖x−y‖X

in a Hilbert spaceY and with orthogonal projectorsπ1(t) leads to (11) with γi(w) = `|w|2 orγi(w) = `|w|1, where| · |1 denotes the sum norm and| · |2 denotes the euclidean norm inR2.

In order to compare known results with ours we verify the assumptions of Theorem3.10 fordifferent forms of Lipschitz estimates forf and for concrete functionsψ in the exponentialdichotomy property.

Corollary 4.1. Under the general assumptions in Sec.3.2, let f satisfy(11) with weighted max-imum norms

γi(w) = `i max{|w1|, |w2|} , `i > 0 for w∈ R2 . (46)

Let t∗, ψ∗ and k5 with the properties as in Theorem3.10. Then condition(29) and hence theclaim of Theorem3.10hold if

β2−β1 >k3`1 +k4k5`2

2+

√(k3`1−k4k5`2)2

4+

k1k2k3k4k25`1`2

ψ∗. (47)

Proof. Calculating the zeroes ofG with

G(ρ) = β2−β1−k3`1max{1,ρ}−k4k5`2ρ−1max{1,ρ} ,

we find (47) as sufficient and necessary condition for (29) and (30).

Latushkin and Layton [LL99] consider−A as generator of a strongly continuous semigroup onthe Banach spaceX = Y. Let X be the direct sum of two subspaceX1 andX2 and letπi theprojector fromX ontoXi . Assuming exponential dichotomy conditions (10) with k1 = k2 = 1(andk3 = k4 = 1, ψ = 1 because ofX = Y) and f (0) = 0 and

‖πi [ f (x)− f (y)]‖X ≤ `i max{‖π1[x−y]‖X,‖π2[x−y]‖X}

for the time-independent nonlinearityf , they found

β2−β1 > `1 + `2 (48)

as optimal spectral gap condition. They extended this result to(−A(t)) as a family of linearoperators on the Banach spaceX = Y generating a strongly continuous semiflow, see [LL99] too.Again, assuming exponential dichotomy conditions (10) with k1 = k2 = k3 = k4 = 1, ψ = 1, andthe Lipschitz estimate

‖πi(t)[ f (t,x)− f (t,y)]‖X ≤ `i max{‖π1(t)[x−y]‖X,‖π2(t)[x−y]‖X}

37

and f (t,0) = 0 for f , they found the spectral gap condition (48) for nonautonomous inertialmanifolds.

SinceX = Y, we havek3 = k1, k4 = k2, ψ = 1. Thus we have to chooset∗ = 0 and findk5 = ψ∗ =1. Our condition (47) reduces to

β2−β1 > k1`1 +k2`2 ,

which in the special case ofk1 = k2 = 1 reduces to the optimal spectral gap condition (48) foundby Y. Latushkin and B. Layton, [LL99].

Corollary 4.2. Under the general assumptions in Sec.3.2, let f satisfy(11) with weighted sumnorms

γi(w) = `i1|w1|+ `i2|w2| , `i1, `i2 > 0 for w∈ R2. (49)

Let t∗, ψ∗ and k5 with the properties as in Theorem3.10. Then condition(29) and hence theclaim of Theorem3.10hold if

β2−β1 > k3`11+k4k5`22+k1k2k5 + ψ∗√

k1k2ψ∗

√`12`21k3k4 . (50)

Proof. Calculating the zeroes ofG with

G(ρ) = β2−β1−k3`11−k3`12ρ−k4k5`21ρ−1−k4k5`22 ,

we find (50) as a sufficient and necessary condition for (29) and (30).

First letψ(t) := max{ααβ−α

2 t−α,1}as in [FST88, Lemma 3.1]. Here and in the following we set 00 := 1 in order to continuouslyextend the expression forψ to the limit caseα = 0. We chooset∗ := αβ−1

2 and hence we haveψ∗ = 1. To satisfy (28) we note that

δ∫ t∗

0ψ(r)e−δr dr + ψ∗ lim

t→t∗e−δt = δαααβ−α

2

∫ δαβ−12

0r−αe−r dr +e−δαβ−1

2 .

The right hand side is monotonously increasing inδ > 0. Therefore, we may satisfy (28) for0< δ≤ β2−β1≤ β2 with

k6 := αα∫ α

0r−αe−r dr +e−α−1≥ 0, k5 := 1+

(β2−β1)α

βα2

k6 .

If k3 = k1βα1 , k4 = k2βα

2 and`11 = `12 = `21 = `22 = `, condition (50) reads

β2−β1 >

(k1βα

1 +k2k5βα2 +(1+k5k1k2)

√βα

1βα2

)` . (51)

38

Now we assume thatY is a Hilbert space,A is a time-independent, selfadjoint, positive linearoperator onY with dense domain and compact resolvent,f is a time-independent, continuousmapping fromX = D(Aα) into Y satisfying a global Lipschitz condition‖ f (x)− f (y)‖Y ≤ `‖x−y‖X. Let λ1 ≤ λ2 ≤ ·· · denote the eigenvalues ofA counted with their multiplicity and letπ1

be the orthogonal projector fromY onto theN-dimensional subspace spanned by the firstNeigenvectors ofA. Then (10) is satisfied withk1 = k2 = 1, β1 = λN, β2 = λN+1, and we find thespectral gap condition

λN+1−λN >

((λα/2

N + λα/2N+1

)2+k6

λα/2N + λα/2

N+1

λα/2N+1

(λN+1−λN)α

)` (52)

which holds ifλN+1−λN > 2

(λα

N + λαN+1 +k6(λN+1−λN)α)` . (53)

Romanov [Rom94] showed that a spectral gap condition

λN+1−λN > (λαN+1 + λα

N)` (54)

is sufficient for the existence of anN-dimesional (autonomous) inertial manifold. Note thatthe right hand side in (53) is at most by the factor 2(1+ k6) worse than the right hand sidein the sharp condition (54), wherek6 = 0 for α = 0 andk6 ≈ 0.46 for α = 1

2. There are tworeasons why our condition (53) is worse than (54): Firstly, Romanov used the Fourier expansionof the points in the Hilbert spaceY and the corresponding spectral properties ofA instead ofthe exponential dichotomy condition. Secondly, he used an indefinite quadratic form along thedifference of trajectories in order to show needed properties of the graph transformation mapping.This approach is more effective in Hilbert spaces than the more general approach presented herefor the verification of the cone invariance and squeezing property. However, using indefinitequadratic forms too, we are also able to show that (54) is sufficient for the cone invariance andsqueezing property.

Now letψ(t) = β−α

2 t−α +1

as in [Tem97]. Thent∗ = ∞, ψ∗ = 1 and we may choose

k6 := Γ(1−α) , k5 := 1+k6

to satisfy (28). In the special case1 = `2 = `, k3 = k1βα1 , k4 = k2βα

2 , condition (50) reads now

β2−β1 >

(k1βα

1 +(1+k1k2(1+k6))√

βα1βα

2 +k2(1+k6)βα2

)` . (55)

For a Banach spaceY, a time-independent, sectorial linear operatorA on Y with dense domainD(A), and a time-independent, continuous mappingf from X = D(Aα) into Y satisfying a global

39

Lipschitz condition‖ f (x)− f (y)‖Y ≤ `‖x− y‖X whereX = D((A + a)α) with fixed a ∈ R,α ∈ [0,1[ with ℜσ(A) + a > 0, and under assumption (10) with ψ(t) = β−α

2 t−α + 1, Temamshowed ([Tem97], Theorem IX.2.1) that there are constantsc1 andc2 independent of the Lips-chitz constant and the boundedness constant`0 of the nonlinearityf , such that the spectral gapcondition

β2−β1≥ c1(`0 + `+ `2)(βα2 + βα

1) , β1−α1 ≥ c2(`0 + `) (56)

implies the existence of an autonomous inertial manifold in the autonomous case.

Note that our condition (55) is of similar form as (56) but (55) contains only known constants andis applicable for the nonautonomous case, too. Moreover, in contrast to (56), in our condition(55), the right hand side is linear in the Lipschitz constant`.

Finally letψ(t) = ααβ−α

2 t−α +1.

Then we chooset∗ := ∞ and haveψ∗ = 1. Since

δ∫ t∗

0ψ(τ)e−δτ dτ = δβ−α

2

∫ ∞

0(αατ−α + βα

2)e−δτ dτ

= ααδαβ−α2 Γ(1−α)+1,

for δ> 0, we may choose

k6 := ααΓ(1−α) , k5 := 1+(β2−β1)α

βα2

k6 (57)

in order to satisfy (28) for δ ∈ ]0,β2−β1[. In the special case1 = `2 = `, k3 = k1βα1 , k4 = k2βα

2 ,condition (50) takes the form (51).

Whilst we are yet not in a position to deal with retardation or stochastic perturbation, we try tocompare our result with that one found by L. Boutet de Monvel, I.D. Chueshov and A.V. Re-zounenko in [BdMCR98] and by I.D. Chueshov, M. Scheutzow in [CS01] for the special case ofa semilinear parabolic equation without perturbation and without retardation. ThereY is a Hilbertspace,A is a time-independent, selfadjoint, positive linear operator onY with dense domain andcompact resolvent,f is a continuous mapping fromR×X, X = D(Aα), intoY satisfying a globalLipschitz condition‖ f (t,x)− f (t,y)‖Y ≤ `‖x− y‖X. Let λ1 ≤ λ2 ≤ ·· · denote the eigenvaluesof A counted with their multiplicity and letπ1 be the orthogonal projector fromY onto theN-dimensional subspace spanned by the firstN eigenvectors ofA. Chueshov and Scheutzow [CS01]found the spectral gap condition

λN+1−λN > 2(λα

N + λαN+1 + ααΓ(1−α)(λN+1−λN)α)` , (58)

and Boutet de Monvel, Chueshov and Rezounenko found

λN+1−λN ≥ 4(λα

N + λαN+1 + ααΓ(1−α)(λN+1−λN)α)` ,

40

which is is little bit worse than (58).

In this situation the exponential dichotomy condition (10) is satisfied withk1 = k2 = 1, β1 = λN,β2 = λN+1, k3 = λα

N, k4 = λαN+1, and we find again the spectral gap condition (52) but here

with k6 given by (57). Obviously our condition (52) is a little weaker than (58). So we havegood chances to extend our result to retarded semilinear parabolic equations and, possibly, tosemilinear parabolic equations with stochastic perturbation.

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