on quadratic integral equation of fractional orders - icm 2006

ICM 2006

Short Communications

Abstracts

Section 11Partial Differential Equations

ICM 2006 – Short Communications. Abstracts. Section 11

On fuzzy integral equation of fractional orders

Mohamed Abdalla Darwish

Department of Mathematics, Faculty of Education, Alexandria University,Damanhour Branch, 22511 Damanhour, [email protected], [email protected]

2000 Mathematics Subject Classification. 45G10

We present an existence and uniqueness theorem for integral equations offractional orders involving fuzzy set valued mappings of a real variablewhose values are normal, convex, upper semicontinuous and compactlysupported fuzzy sets in Rn. The method of successive approximation isthe main tool in our analysis.

References

[1] Darwish, M.A., On Quadratic Integral Equation of Fractional Orders, J. Math.Anal. Appl. 311 (2005), 112–119.

[2] Dubois, D., Parde, H., Towards Fuzzy Differential Calculus, Part 1. Fuzzy Setsand Systems 8 (1982), 1–17.

[3] Dubois, D., Parde, H., Towards Fuzzy Differential Calculus, Part 2. Fuzzy Setsand Systems 8 (1982), 105–116.

[4] Friedman, M., Ming, Ma, Kandel, A., Solutions to Fuzzy Intergal Equationswith Arbitrary Kernels, Internat. J. Approx. Reason. 20 (3) (1999), 249–262.

ICM 2006 – Madrid, 22-30 August 2006 1


Spectral asymptotics and matrix geometry

Ivan G. Avramidi

Department of Mathematics, New Mexico Institute of Mining and Technology,Socorro, NM 87801, [email protected]

2000 Mathematics Subject Classification. 58J50, 58J60

We study second-order elliptic partial differential operators acting on sec-tions of vector bundles over a compact manifold with a non-scalar positivedefinite leading symbol. Such operators, called non-Laplace type operators,appear, in particular, in gauge field theories, string theory as well as mod-els of non-commutative gravity theories. We show that the leading symbolof such an operator naturally defines a collection of Finsler geometries onthe manifold, which can be thought of as a non-commutative deformationof Riemannian geometry when instead of a Riemannian metric there isa matrix valued self-adjoint symmetric two-tensor that plays the role ofa “non-commutative” metric. It is well known that there is a small-timeasymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time, with the coefficients being the spectral invariantsof the operator. We initiate the development of a systematic approach forthe explicit calculation of the coefficients of this asymptotic expansion. Wecompute explicitly the first two heat trace coefficients for manifolds withboundary and the first three coefficients for manifolds without boundary.We propose a non-commutative deformation of the Einstein-Hilbert actionfunctional as a linear combination of the first three spectral invariants. Thecritical points of this functional naturally define the “non-commutative”generalization of Einstein equations.

References

[1] Avramidi, I. G., Non-Laplace type Operators on Manifolds with Boundary. InAnalysis, Geometry and Topology of Elliptic Operators, Papers in Honor ofKrzysztof Wojciechowski (ed. by B. Booss-Bavnbek, S. Klimek, M. Lesch andW. Zhang). World Scientific, Singapore, 2006, pp. 119–152.

[2] Avramidi. I. G., Dirac Operator in Matrix Geometry, Int. J. Geom. Meth. Mod.Phys., 2 (2005), 227–264.

[3] Avramidi, I. G., Gauged Gravity via Spectral Asymptotics of non-Laplace typeOperators, J. High Energy Phys., 07 (2004), 030.

2 ICM 2006 – Madrid, 22-30 August 2006


[4] Avramidi, I. G., Matrix General Relativity: A new look at Old Problems, Class.Quant. Grav., 21 (2004), 103–120.

[5] Avramidi, I. G., A Noncommutative Deformation of General Relativity, Phys.Lett., B576 (2003), 195–198.



Phase transitions in linear dynamical systems with applications

Jacek Banasiak

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041,South [email protected]

2000 Mathematics Subject Classification. 47D06

Mathematical models of many physical and biological systems do not re-turn conservation laws used to derive them. For instance, fragmentationand coagulation models, derived from the principle of conservation of massmay have solutions which are not mass conserving. Such phenomena arecalled ’gelation’ (in coagulation) and ’shattering’ (in fragmentation), andare attributed to a phase transition creating a state which is beyond theresolution of the model. Some processes of this type can be modelled asMarkov processes and, if such a phase transition occurs, they are referredto as explosive Markov processes. An analytic theory of such processes, de-veloped within the last several years, covers a wider range of applications,and has provided an exhaustive characterization of cases when such phasetransitions occurs in terms of the properties of the generator of the under-lying dynamical system, see [1]. We present the recent results of this theorytogether with applications to models arising in biology, kinetic theory andother areas of physical sciences.

References

[1] J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications,Springer Monographs in Mathematics, Springer, London, 2006.



Linear stability of a self-gravitating compressible fluid with afree boundary

Uwe Brauer*, Gabriel Nagy

Departamento de Matematica Aplicada, Universidad Complutense de Madrid,Spain; Department of Mathematics, University of California at San Diego, [email protected]

2000 Mathematics Subject Classification. 76N10, 35L50, 35Q35

We consider the initial free–boundary value problem for the self-gravitatingcompressible three dimensional Euler equations with positive mass densityat the boundary, for which we prove the linear stability of static backgroundsolutions. Our work can be summarised in two essential steps. First, wetransform the free–boundary problem into a fixed-boundary problem in thecommon way by using the Lagrange formulation of Euler’s equations. Wethen write the resulting system as a first order system of evolution andconstraint equations. Second, we enlarge the system including every firstderivative of the fluid velocity as a new variable. This procedure leads toawhole class of systems with different evolution equations. One of thesesystems admits a symmetric hyperbolic formulation of the evolution equa-tions which might be useful for numerical investigations. Another of thesesystems allows to decouple certain evolution equations, which can then besolved independently. We prove well posedness for the linearization of theseequations near a static background. This is done using known results forthe initial fixed-boundary value problem for linear symmetric hyperbolicsystems. The treatment of the constraints at the boundary turns out to bethe most difficult part of our approach.

References

[1] T. Beale, T. Hou, and J. Lowengrub, Growth rates for the linearizedmotion of fluid interfaces away from equilibrium, Communications on Pureand Applied Mathematics, 46 (1993), p. 1269.

[2] H. Friedrich, Evolution equations for gravitating ideal fluid bodies in generalrelativity, Phys. Rev. D, 57 (1998), pp. 2317–2322.

[3] H. Lindblad, Well posedness for the linearized motion of an incompress-ible liquid with free surface boundary, Commun. Pure Appl. Math., 56 (2003),pp. 156–197.

[4] P. Secchi, The initial-boundary value problem for linear symmetric hyper-bolic systems with characteristic boundary of constant multiplicity, DifferentialIntegral Equations, 9 (1996), pp. 671–700.



Some asymptotic properties for convection-diffusion equations

Pablo Braz e Silva1*, Paulo Zingano2

1 Department of Mathematics, Universidade Federal de Pernambuco, 50740-540,Recife, Pernambuco, [email protected]

2 Departament of Pure and Applied Mathematics, Universidade Federal do RioGrande do Sul, 91509-900, Porto Alegre, RS, [email protected]

2000 Mathematics Subject Classification. 35B40, 35K15, 35K55

We discuss some asymptotic properties for solutions of general convection-diffusion equations

ut + divf(x, t, u) = div(A(x, t, u)∇u), x ∈ Rn, t > 0,

with initial data u(x, 0) = u0(x) ∈ Lp(Rn), 1 ≤ p < ∞. Under suitableassumptions on f and A, we show

‖u(·, t)‖L2p ≤ C(n, p)‖u0‖Lp(tµ(t))−n4p , (1)

where µ(t) is a positive non-increasing function such that 〈A(x, t, ξ)ξ, ξ〉 ≥µ(t)|ξ|2, ∀ ξ ∈ Rn. Moreover, we discuss how the bounds (2) can be used toobtain an L∞ estimate

‖u(·, t)‖∞ ≤ K(n, p)‖u0‖p(tµ(t)

)− n2p .

Some interesting applications of these properties are also mentioned. Bounds(2) are similar to the ones established in [1] for the one-dimensional case.

References

[1] Braz e Silva, P., Zingano, P., Some asymptotic properties for solutions of one-dimensional advection-diffusion equations with Cauchy data in Lp(R), C. R.Acad. Sci. Paris, Ser. I, Math. 342 No 7 (2006), 465–467



Asymptotic behaviour of the porous media equation in domainswith holes

Cristina Brandle*, Fernando Quiros, Juan Luis Vazquez

Departamento de Matematicas, U. Autonoma de Madrid, 28049 Madrid, [email protected]

2000 Mathematics Subject Classification. 35B40, 35R35, 35K65

Let G ⊂ RN be a bounded open set with smooth boundary and let Ω =RN \G. We study the large-time behaviour of the solution to

ut = ∆um, (x, t) ∈ Ω× (0,∞),u(x, t) = 0, (x, t) ∈ ∂Ω× (0,∞),u(x, 0) = u0(x), x ∈ Ω,

(1)

where m > 1. We assume u0 to be in L1(Ω), nonnegative in Ω and compactlysupported in Ω. We restrict to N > 2, since the situation for N = 1, 2 isdifferent.

As a first step we show that u decays as O(t−α), while its supportexpands like O(tβ), where α = N/(N(m − 1) + 2) and β = 1/(N(m −1) + 2). We scale u according to these rates, vout(y, τ) = tαu(ytβ , t), τ =log t, and prove that vout converges as τ → ∞ to the profile FC? of aparticular Barenblatt solution, BC?(x, t) = t−αFC?(xt−β). The constant C?

is determined from the initial data thanks to an explicit conservation lawthat takes into account the amount of mass lost through the boundary.This amount is given by the projection of the initial data on a function Φwhich is the normalized harmonic function that measures the capacity ofG. Convergence is uniform in sets |y| ≥ δ, i.e., in a wide exterior regionup to the free boundary, what is called in Matched Asymptotics the outerlimit.

To complete the study we consider what happens in the region nearthe holes (so-called inner limit). In this case we only have to amplify thesolution, keeping the space variable fixed (i.e., a quasi-stationary situation).We prove that the rescaled function vin(x, t) = tαu(x, t) converges to astationary state, HC(x) = C

mm−1H(x), where H = (1 − Φ) is the unique

solution of

∆H = 0, x ∈ Ω, H = 0, x ∈ ∂Ω, H → 1, |x| → ∞. (2)

The free constant C is adjusted through matching with the Barenblattfunction which gives the outer behaviour. It turns out that C = C?. Com-bining the inner and outer descriptions allows us to write a global uniformapproximation for the large-time behaviour of the solution.



The asymptotic behaviour of the solution to (1) was studied formallyby King in the radial case [1]. Very recently Gilding and Goncerzewicz [2]have performed a rigorous outer analysis by completely different methods.

References

[1] King, J. R., Integral results for nonlinear diffusion equations, J. Engrg. Math.25 (1991), no. 2, 191–205.

[2] Gilding, B. H., Goncerzewicz, J., Large-time behaviour of solutions of theexterior-domain cauchy-dirichlet problem for the porous media euqation withhomogeneous boundary data, Monatsh. Math, to appear.



Homogenization of parabolic problems with varying boundaryconditions

Carmen Calvo-Jurado*, Juan Casado-Dıaz, Manuel Luna-Laynez

Department of Mathematics, University of Extremadura, Spain; Department ofDifferential Equations and Numerical Analysis, University of Sevilla, [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 35B40

Given Ω a Lipschitz bounded open set of RN , N ≥ 2, and Γn an arbitrarysequence of subsets of ∂Ω, our purpose in this work is to study the asymp-totic behavior of the solution un of the following linear parabolic problem

∂tun − div (A∇un −Gn) = fn in Ω× (0, T )un = 0 on Γn × (0, T ), (A∇un −Gn) ν = 0 on (∂Ω \ Γn)× (0, T )un(x, 0) = 0 in Ω,

(1)with A ∈ L∞(Ω× (0, T ))N×N elliptic and fn ∈ L2(Ω× (0, T )), Gn ∈ L2(Ω×(0, T ))N such that fn converges weakly in L2(Ω × (0, T )) to a function fand Gn converges strongly in L2(Ω× (0, T ))N to a function G.

Under these conditions, we show the existence of a subsequence of n,still denoted by n, a nonnegative Borel measure µ which vanishes on thesets of capacity zero and a positive function b ∈ L∞µ (Ω× (0, T )), such thatfor every sequences fn and Gn as above, the solution un of (1) convergesweakly in L2(0, T,H1(Ω)) and strongly in L2(0, T,H1

loc(Ω)) to the uniquesolution u of the variational problemu ∈ L2(0, T ;H1(Ω) ∩ L2

µ(∂Ω)), u(x, 0) = 0 in Ω∫Ω∂tuvdx+

∫ΩA∇u∇vdx+

∫∂Ωbuvdµ =

∫Ωfvdx+

∫ΩG∇vdx in D′(0, T )

∀v ∈ H1(Ω) ∩ L2µ(∂Ω).

We observe that assuming smoothness enough, the boundary condition sat-isfied by u is(A∇u−G) ν + buµ = 0 on ∂Ω.

References

[1] C. Calvo-Jurado, J. Casado-Dıaz, Asymptotic behaviour of linear Dirichletparabolic problems with variable operators depending on time in varying do-mains. In J. Math. Ann. and Appl. 306 (2005), 282–316.



[2] C. Calvo-Jurado, J. Casado-Dıaz, M. Luna-Laynez, Homogenization of ellip-tic problems with the Dirichlet and Neumann conditions imposed on varyingsubsets. Submitted for publication.

[3] D. Cionarescu, F. Murat, Un terme etrange venu d’ailleurs. In Nonlinear partialdifferential equations and their applications. College de France seminar, VolsII and III. Eds H. Brezis and J. L. Lions. Research Notes in Math. 60 and 70.Pitman, London, (1982) 98–138 and 154–78.

[4] G. Dal Maso, A. Garroni. New results on the asymptotic behaviour of Dirichletproblems in perforated domains. In Math. Models Methods Appl. Sci. 3 (1994),373-407.



Existence of solutions for a dynamic Signorini’s contact problem

M. T. Cao*, P. Quintela

Departamento de Matematica Aplicada, Universidade de Santiago de Compostela,Santiago de Compostela, 15782, [email protected], [email protected]

2000 Mathematics Subject Classification. 74B10, 74H20, 74M15, 35D05,35K85

The purpose of this talk is to present an existence result for the dynamicfrictionless contact problem between an elastic body and a rigid founda-tion. The problem of the wave equation with unilateral boundary condi-tions has been studied by Lebeau and Schatzman [4] who established theexistence and uniqueness of solution for half-space domains. As the authorsthemselves explain, however, their method cannot be extended to generaldomains. Smooth bounded domains have been considered by Kim in [2] buthis work cannot be applied to elasticity.In order to model the contact we consider Signorini conditions. So, the con-tact problem can be posed as follows:Problem (P) Find (u,σ) verifying:

ρu− divσ(u) = F in Ω× (0, T ), (1)σn = g on ΓN × (0, T ), (2)

u = 0 on ΓD × (0, T ), (3)σt = 0; σn ≤ 0; un ≤ 0; σnun = 0 on ΓC × (0, T ), (4)

u(x, 0) = u0; u(x, 0) = u1 in Ω, (5)

where ρ is the density of the solid, Ω ⊂ Rn, n = 2, 3 is a bounded domain ofclass C1,1, g ∈W 2,∞(0, T ; [L2(ΓN )]n∩[H− 1

2 (Γ)]n), F ∈W 2,∞(0, T ; [L2(Ω)]n),σ(u) = Λ−1ε(u), Λ being the elasticity tensor assumed to be time indepen-dent, symmetric and coercive, n denotes the unit outward normal vectorand mes(ΓD) > 0. The initial conditions u0 and u1 are assumed to belongto [H1(Ω)]n such that divΛ−1ε(u0) ∈ [L2(Ω)]n.The proof is based on five fundamental steps: a discretization in time, usingNewmark’s method, which leads to a discretized problem with unique solu-tion; the construction of functions approximating a solution of the problem;the treatment of the contact condition by means of a Lagrange multiplierwhose orthogonality properties allow us to get a priori estimates; the con-vergence of said functions and, finally, the pass to the limit obtaining aweak solution of the continuous problem.



References

[1] P. Barral and P. Quintela, Existence of a solution for a Signorini contact prob-lem for Maxwell-Norton materials, IMA Journal of Applied Mathematics 67(2002) 525–549.

[2] J.U. Kim, A boundary thin obstacle problem for a wave equation, Commun.in Partial Differential Equations 14 (1989) 1011–1026.

[3] J.L. Lions, Quelques methodes de resolution des problemes aux limites nonlineaires (Dunod, 1969).

[4] G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateralconstraint at the boundary, J. Diff. Eqs. 53 (1984) 309–361.



A new method for bounded and blowup solutions of parabolicequations

Shaohua Chen

Department of Mathematics, Cape Breton University, Sydney, NS, B1P 6L2,Canadageorge [email protected]

2000 Mathematics Subject Classification. 35K20, 35K55, 35K65

We introduce a new method to obtain an a priori estimates, that is, wedifferentiate the integration of a ratio of two solutions. Several kinds ofequations can use this method.

First, we consider the quasilinear parabolic equation ut = ∇·(g(u)∇u)+h(u,∇u)+f(u) with u|∂Ω = 0, u(x, 0) = φ(x). If f, g and h are polynomialswith proper degrees and proper coefficients, we will show that the blowupproperty only depends on the first eigenvalue λ1 of −∆ in Ω with Dirichletboundary condition. For the special case, ut = ∇ · (uα∇u) + c1u

α−1|∇u|2 +c2u

α+1 with α > 0 and c2 > 0, our conclusion is: if and only if c2(1+α+c1) >λ1, then, for any initial value φ(x) ∈ C1+β

0 (Ω) with β > 0, the solution blowsup in a finite time. The results generalize or complement many previousconclusions in the literature (see [1] and [2]).

Next, we discuss a generalized activator-inhibitor model ut = ε∆u−µu+up/vq and vt = D∆v−νv+ur/vs, with the Neumann boundary conditions.In general, the problem is quite difficult to solve because it has neither avariational structure nor a priori estimates and the comparison principleis failed. By using the new method, we can prove that if p − 1 < r andrq > (p−1)(s+1) then for any positive initial values, the positive solutionsexist for all time (Jiang [2] also used the similar method to obtain thisresult). If we add the additional condition q < s+ 1, then the solutions areLα bounded for any α. We also discuss the uniform bound for the stationarysolutions.

Then, we deal with the heat equations (ui)t =∆ui+f(u1, · · · , um), ui(x, t)|∂Ω =0, ui(x, 0) = φi(x) for i = 1, · · · ,m. Under suitable and natural conditions,the solutions blow up in a finite time. Numerical computations show thatthe conditions couldn’t be weaken.

References

[1] J. Ding, Blow-up solutions for a class of nonlinear parabolic equations withDirichlet boundary conditions, Nonlinear Anal. 52 (2003), 1645-1654.



[2] H. Chen, Analysis of blowup for a nonlinear degeneate parabolic equation, J.Math. Anal. Appl. 192 (1995), 180-193.

[3] Huiqiang Jiang, Global existence of solutions of an activator-inhibitor system,Discrete Contin. Dyn. Syst. 14 (2006), 737–751.



The mean curvature equation in pseudohermitian geometry

Jih-Hsin Cheng

Institute of Mathematics, Academia Sinica, Taipei, R.O.C. (Taiwan)[email protected]

2000 Mathematics Subject Classification. 35L80, 35J70, 32V20, 53A10,49Q10

I will make a brief report on the recent study about the mean curvatureequation in pseudohermitian geometry ([1], [2]). As a differential equation,this (p-)minimal surface equation is degenerate (hyperbolic and elliptic) indimension 2 while subelliptic in the nonsingular domain for higher dimen-sions. We analyze the singular set and formulate an extension theorem.This allows us to classify the entire C2-smooth solutions to this equationand to solve a Bernstein-type problem. As a geometric application, we provethe nonexistence of C2-smooth hyperbolic surfaces having bounded p-meancurvature, immersed in a pseudohermitian 3-manifold. From the variationalformulation of the equation, we study the Dirichlet problem by proving theexistence and the uniqueness of the (p-)minimizers ([3]).

References

[1] Cheng, J.-H., Hwang, J.-F., Malchiodi, A., and Yang, P., Minimal surfacesin pseudohermitian geometry, Annali della Scuola Normale Superiore di Pisa,Classe di Scienze (5), 4 (2005), 129–177.

[2] Cheng, J.-H. and Hwang, J.-F., Properly embedded and immersed minimalsurfaces in the Heisenberg group, Bull. Aus. Math. Soc., 70 (2004), 507–520.

[3] Cheng, J.-H., Hwang, J.-F., and Yang, P., Existence and uniqueness for p-areaminimizers in the Heisenberg group, preprint 2006, arXiv: math.DG/0601208.



Free vibrations for an asymmetric beam equation

Q-Heung Choi*, Tacksun Jung, Hyewon Nam

Department of Mathematics Education, Inha University, Incheon 402-751, Korea;Department of Mathematics, Kunsan National University, Kunsan 573-360,Koreaqheung@@inha.ac.kr; tsjung@@kunsan.ac.kr

2000 Mathematics Subject Classification. 35Q40, 35Q80

We consider the semilinear beam equation where the nonlinear term is afunctions with some powers:

utt + uxxxx + bu+ = f(x, t, u) in (−π2,π

2)×R,

u(±π2, t) = uxx(±π

2, t) = 0, (1)

u is π-periodic in t and even in x and t,

where u+ = maxu, 0 and f is defined by

f(x, t, s) =

|s|p−2s, amp; s ≥ 0|s|q−2s, amp; s lt; 0

(2)

where p, q ≥ 2 and p 6= q.McKenna and Walter proved that if 3 < b < 15 then at least two π-

periodic solutions exist, one of which is large in amplitude. The existence ofat least three solutions was later proved by Choi, Jung and McKenna , usinga variational reduction method. Humphreys proved that there exists an ε >0 such that when 15 < b < 15+ε at least four periodic solutions exist. Choiand Jung suppose that 3 < b < 15 and f is generated by eigenfunctions.Since Micheletti and Saccon applied the limit relative category to studyingmultiple nontrivial solutions for a floating beam.

In this talk, we use a variational approach and look for critical points ofa suitable functional I on a Hilbert space H. Since the functional is stronglyindefinite, it is convenient to use the notion of limit relative category.



Standing waves of some coupled nonlinear Schrodinger equations

A. Ambrosetti, E. Colorado*

Department of Mathematical Analysis, Faculty of Sciences, University ofGranada, 18071-Granada, [email protected], [email protected]

2000 Mathematics Subject Classification. 34A34, 34G20, 35Q55

In spite of the interest that system of coupled NLS (Nonlinear Schrodinger)equations have in Nonlinear Optics, see e.g. [1], only few rigorous generalresults have been proved so far. Here, motivated by the recent paper [4], wedeal with the system

−∆u+ λ1u = µ1u3 + βuv2, u ∈W 1,2(RN )

−∆v + λ2v = µ2v3 + βu2v, v ∈W 1,2(RN )

(1)

where λi, µi > 0, i = 1, 2, β is a real parameter and x ∈ RN , N = 2, 3.We prove the existence of bound and ground states provided the coupling

parameter β < Λ, respectively, β > Λ′, where 0 < Λ ≤ Λ′ < ∞. The mainresults are the following.

Theorem 1. If β > Λ′ then (1) has a (positive) radial ground state u.

Theorem 2. If β < Λ, then (1) has a radial bound state u∗ such thatu∗ 6= uj , j = 1, 2, where u1 = (U1, 0), u2 = (0, U2) and Uj is the positiveradial solution to −∆Uj + λjUj = µjU

3j . Furthermore, if β ∈ (0,Λ), then

u∗ > 0.

The main idea in the proof of Theorems 1, 2 is to show that the Morseindex of u1 and u2 changes with β:

• for β < Λ small their index is 1,

• while for β > Λ′ their index is greater or equal than 2.

This fact, jointly with an appropriate use of the natural constraint method,allows us to prove the existence of bound and ground states.

These results are announced in [2] and proved in [3].

References

[1] N. Akhmediev & A. Ankiewicz, “Solitons, nonlinear pulses and beams”,Champman & Hall, London, 1997.



[2] A. Ambrosetti & E. Colorado, Bound and ground states of coupled nonlinearSchrodinger equations, C. R. Math. Acad. Sci. Paris 342 (2006), no. 2, 453-458.

[3] A. Ambrosetti & E. Colorado, Standing Waves of Some Coupled NonlinearSchrodinger Equations, Preprint SISSA 2006.

[4] T-C. Lin & J. Wei, Ground state of N coupled nonlinear Schrodinger equationsin RN , N ≤ 3, Comm. Math. Phys. 255 (2005), 629-653.



Regularization method for a free boundary problem of parabolictype modelling the barbotage phenomenon in a molten metal

Constantin Ghita

Department of Mathematics, Valahia University of Targoviste, [email protected]

We define a piezometric capacity of a saturated zone during a barbotageprocess and we formulate a Stefan problem; we also derive the theoreticalconsequences of the directional evolution of the purificator fluid front. Thedirectional barbotage can be expressed as a unilateral problem and allowus to split the unknown function in two components: the directional andresidual piezometric capacities, which permit a control of front evolution insaturated zone and formulation of a variational inequality with initial data.We establish by regularization techniques the existence and uniqueness re-sults. In addition, some regularity conditions upon initial data assure us alocal / global identification of weak solution with the classical one for thefree boundary problem.

References

[1] Magenes,E., Topics in parabolic equations, Some typical problems for linearevolution partial equations, In Boundary Value Problems for Linear EvolutionPartial Equations (ed. by Garnir), 1977, 239-312.

[2] Ghita, C. Mathematical metallurgy,A non-standard analysis of metallurgicalprocesses (in romanian). Romanian Academy Press, Bucharest, 1995.



Standing waves for a generalized Davey-Stewartson system

A. Eden*, S. Erbay

Department of Mathematics, Bogazici University, 34342 Bebek-Istanbul, Turkey;Department of Mathematics, Isik University, 34980 Sile-Istanbul, [email protected]; [[email protected]

2000 Mathematics Subject Classification. 35J20, 35Q55

A generalized Davey-Stewartson (GDS) system involving three equationswas derived in [1], and the study of its qualitative properties was initiatedin [2]. Here, we consider the existence of standing waves of GDS system inthe purely elliptic case. The relevant equation is

∆R− ωR− χR3 − bK(R2)R = 0,

where K(f)(ξ) = α(ξ)f(ξ) and

α(ξ) =λξ41 + (1 +m1 − 2n)ξ21ξ

22 +m2ξ

42

(λξ21 +m2ξ22)(ξ21 +m1ξ22).

Our main result establishes the existence of solutions for the above equa-tion, hence of the standing waves, under some conditions on the physicalparameters χ, b,m1 and ω. We also establish an alternative sufficient con-dition for the global existence of solutions to the initial value problem forthe GDS system.

References

[1] Babaoglu, C. and Erbay, S., Two-dimensional wave packets in an elastic solidwith couple stresses, Int. J. Non-Linear Mech. 39 (2004), 941–949.

[2] Babaoglu, C., Eden, A. and Erbay, S., Global existence and nonexistence resultsfor a generalized Davey-Stewarson system, J. Phys. A: Math. Gen. 37 (2004),11531–11546.



Optimal BMO estimates near the boundary for solutions ofscalar and systems of elliptic problems

Azzeddine El Baraka

Department of Mathematics, University Sidi Mohamed Ben Abdellah, FST deFez, BP 2202, Route Immouzer, 30000 Fez, [email protected]

2000 Mathematics Subject Classification. 35J40

We give the optimal a priori estimates for solutions of scalar regular ellipticboundary value problems, in the general Lp,λ,s (Ω) spaces which containthe classical spaces of John and Nirenberg BMO, Campanato spaces L2,λ

and their local versions bmo and l2,λ as special cases. This work improvesa result of Campanato [1] where he got the estimates for the gradient ofsolutions of such problems, and it solves a problem raised by Triebel in [4,section 4.3.4]. In the second part we show that this method (used for thescalar case) works for elliptic systems in the sense of Douglis and Nirenberg,and we extend both the scalar case work of the author [3] and the work ofCampanato [2] where he obtained the regularity for the gradient of solutionsof second order linear strongly elliptic systems.

References

[1] S. Campanato, Equazioni ellipttiche del II ordine e spazi L2,λ, Ann. math.pura. Appl., 69, (1965).

[2] S. Campanato, Sistemi ellitici in forma divergenza. Regolarita all’interno,Quaderni Sc. Norm. Sup. Pisa, (1980).

[3] A. El Baraka, Optimal BMO and Lp,λ estimates for solutions of elliptic bound-ary value problems, Arab. J. Sci. Engeneering, Vol. 30, Number 1A, (2005).

[4] H. Triebel, Theory of function spaces, Birkhauser, (1983).



Two remarks on a generalized Davey-Stewartson system

A. Eden, H. A. Erbay*, G. M. Muslu

Department of Mathematics, Bogazici University, Bebek 34342, Istanbul, Turkey;Department of Mathematics, Isik University, Sile Campus, 34980 Sile, Istanbul,Turkey; Department of Mathematics, Istanbul Tech. University, Maslak 34469,Istanbul, [email protected]

2000 Mathematics Subject Classification. 35Q55

Many equations can be expressed as a cubic nonlinear Schrodinger (NLS)equation with additional terms, such as the Davey-Stewartson (DS) system[1]. As it is the case for the NLS equation, the solutions of the DS systemare invariant under the pseudo-conformal transformation. For the ellipticNLS, this invariance plays a key role in understanding the blow-up profileof solutions, whereas in the hyperbolic-elliptic case of DS system an explicitblow-up profile is obtained via the pseudo-conformal invariance. An anal-ogous system has been derived in [2] to model wave propagation in a gen-eralized elastic medium and has been called Generalized Davey-Stewartson(GDS) system. In [3], for the hyperbolic-elliptic-elliptic and elliptic-elliptic-elliptic cases the GDS system has been expressed as a NLS equation withnon-local terms.

We present two results on the GDS system, both following from thepseudo-conformal invariance of its solutions. In the hyperbolic-elliptic-ellipticcase, under some conditions on the physical parameters, we establish ablow-up profile. These conditions turn out to be necessary conditions forthe existence of a special “radial” solution. In the elliptic-elliptic-ellipticcase, under milder conditions, we show the Lp-norms of the solutions decayto zero algebraically in time for 2 < p <∞.

References

[1] Ghidaglia, J. M., Saut, J. C., On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3 (1990) 475-506.

[2] Babaoglu, C., Erbay, S., Two dimensional wave packets in an elastic solid withcouple stresses, Int. J. Nonlinear Mech. 39 (2004) 941-949.

[3] Babaoglu, C., Eden, A., Erbay, S., Global existence and nonexistence resultsfor a generalized Davey-Stewartson equation, J. Phys. A-Math. Gen. 37 (2004)11531-11546.



Initial boundary value problems for the KdV equation infractional-order Sobolev spaces

Andrei V. Faminskii

Peoples’ Friendship University of Russia, Miklukho–Maklai str. 6, Moscow,117198, [email protected]

2000 Mathematics Subject Classification. 35Q

Global well–posedness is established for three initial boundary value prob-lems for the Korteweg – de Vries equation

ut + uxxx + aux + uux = f(t, x)

in 1) a right half–strip Π+T = (0, T )×R+, 2) a left half–strip Π−

T = (0, T )×R−and 3) a bounded rectangle QT = (0, T ) × (0, 1) (T > 0 – arbitrary).Besides an initial condition u(0, x) = u0(x) boundary conditions are set,which are different for the considered problems: u(t, 0) = u1(t) for the firstone, u(t, 0) = u2(t), ux(t, 0) = u3(t) for the second one and u(t, 0) = u1(t),u(t, 1) = u2(t), ux(t, 1) = u3(t) for the third one.

It is assumed, that the initial data u0 ∈ Hs, where s ≥ 0 and s 6= 3k +1/2, k ≥ 0 – integer, for all three problems and, in addition, s 6= 3k+3/2 forthe latter two ones. The boundary data u1, u2 ∈ H(s+1)/3+ε, u3 ∈ Hs/3+ε,where ε = 0 if s ≥ 1 for the problem in Π+

T , if s ≥ 2 for the problem in Π−T ,

if s ≥ 3 for the problem in QT , and ε > 0 is arbitrary small for the lessvalues of s. Such conditions are natural (or ε–close to natural) in the sense,that they originate from properties of solutions to a corresponding initialvalue problem for a linearized KdV equation vt + vxxx = 0.

Solutions of the considered problems are constructed in special func-tional spaces of Bourgain type. They also preserve Hs class with respect tox for every t ∈ [0, T ] and possess a certain property of extra local smoothing.

The work was supported by RFBR grant 06–01–00253.

References

[1] Faminskii, A. V., An Initial Boundary–Value Problem in a Half–Strip for theKorteweg – de Vries Equation in Fractional–Order Sobolev Spaces, Comm.Partial Differential Equations 29 (2004), 1653–1695.

[2] Faminskii, A. V., On Two Initial Boundary Value Problems for the GeneralizedKdV Equation, Nonlinear Boundary Problems 14 (2004), 58–71.



Maximal regularity for Kolmogorov operators in L2 spaces withrespect to invariant measures

Balint Farkas*, Alessandra Lunardi

Technische Universitat Darmstadt, Department of Applied Analysis,Schlossgartenstrasse 7, 64289 Darmstadt, [email protected]

2000 Mathematics Subject Classification. 35H10, 35J15, 35B65, 35J70,47D06, 47N20, 46B70

We are concerned with the differential operator

Lu(x) =12

d∑i,j=1

qijDiju(x) +d∑

i,j=1

bijxjDiu, x ∈ Rd,

where B = (bij) and Q = (qij) are real d × d-matrices, Q is symmetricand nonnegative. Therefore L is a possibly degenerate elliptic operator thatwe assume to be hypoelliptic, and that is called Kolmogorov or degenerateOrnstein–Uhlenbeck operator. The hypoellipticity assumption here is thefollowing: the symmetric matrices Qt defined by Qt :=

∫ t0 e

sBQesB∗ds have

nonzero determinant for all t > 0. This implies that the Gaussian measuresNetBx,Qt

with covariance operator Qt and mean etBx (t > 0, x ∈ Rd) are allabsolutely continuous with respect to the d-dimensional Lebesgue measure.With the aid of such measures the Ornstein–Uhlenbeck semigroup (T (t))t≥0

is readily defined by (T (t)f)(x) =∫Rd fdNetBx,Qt

.Together with hypoellipticity, the other structural assumption is ex-

istence of an invariant measure for L, i.e., a probability measure µ suchthat

∫Rd Ludµ = 0 for all u ∈ C2

b (Rd). The Ornstein–Uhlenbeck semigroupis then strongly continuous on L2(Rd, µ) and its infinitesimal generator(L,D(L)) is a “realization” of the above operator L.

Our main theorem states that the domain of L is continuously embeddedin the fractional weighted anisotropic Sobolev spaceH2,2/3,...,2/(2n−1)(Rd, µ).Here the anisotropic splitting of the space comes from the hypoellipticityassumption.

Since our weighted Lebesgue and Sobolev spaces are locally equivalentto the usual Lebesgue and Sobolev spaces, it follows that for each u ∈D(L) there exist the derivatives Diu, Diju for i, j ∈ I0 (I0 is the firstgroup of variables in the above “ansiotropic decomposition”) and they arein L2

loc(Rd,dx); moreover u ∈ H2/(2n−1)loc (Rd,dx). The last exponent 2/(2n−

1) agrees with the general local regularity results of [2]. Concerning localmaximal regularity, we mention also the paper [1] where it was proved thatthe second order derivatives Diju, i, j ∈ I0, exist and belong to L2

loc(Rd,dx).



References

[1] G. B. Folland Subelliptic estimates and function spaces on nilpotent Lie groups,Ark. Mat. 13 (1975), no. 2, 161–207.

[2] L. P. Rothschild, E. M. Stein Hypoelliptic differential operators and nilpotentgroups, Acta Math. 137 (1976), no. 3-4, 247–320.



Some global differentiability results for solutions of nonlinearelliptic problems with controlled growths (work in progress)

Luisa FattorussoDIMET. Facolta di Ingegneria Universita ”Mediterranea” degli Studi di ReggioCalabria. Via Graziella (Loc.Feo Di Vito)- Reggio Cal., [email protected]

2000 Mathematics Subject Classification. 35J60, 35D10, 58B10

Let Ω be a bounded open subset of Rn n > 2.In Ω we deduce the globaldifferentiability result

u ∈ H2(Ω, RN )

for the solution u ∈ H1(Ω, RN ) of the Dirichlet problem:

u− g ∈ H10 (Ω, RN ) (1)

−∑

iDiai(x, u,Du) = B0(x, u,Du)

with controlled growths and non linearity q = 2.In particular we assumethat ∀(x, u, p) ∈ Λ = Ω×RN ×RnN with |u| ≤ k

‖B0(x, u, p)‖ ≤ f0(x) + c(k)‖u‖α +∑j

‖pj‖γ

where f0 ∈ L2(Ω), α ≤ nn−2 γ ≤ n+2

n .In the first part of this paper we obtain this result in a particular case(γ = 1) of this controlled growths, taking into account a local differen-tiability result proved by S.Campanato, achieving later a differentiabilitytheorem near the boundary and then the global differentiability result us-ing covering procedure.Afterwards we obtain a more general differentiabilityresult (γ ≤ n+2

n ) first proving a local differentiability result for a solutionu ∈ H1(Ω, RN )∩C0,λ(Ω, RN ) of the problem (1) making use of interpolationtheorem in Besov’s spaces and embedding theorem of Gagliardo-Nirembergtype for functions u ∈Wm,r ∩Cs,λ; proceeding analogously to the first partwe obtain the global differentiability result.

References

[1] M.Marino-A.Maugeri, Differentiability of wake solution of non linear parabolicsistems with quadratic growts, Le Matematiche Vol L ,(1995) Fasc. II , 371-377.

[2] S. Campanato, Sistemi ellittici in forma di divergenza, Quad. Scuola Norm.Sup. Pisa (1980). zb10453.35026

[3] L. Fattorusso, Sulla differenziabilita di sistemi parabolici non lineri del II ordinead andamento quadratico, Bollettino UMI (7) 1B (1987),741-764.



Behavior of solutions of the first mixed problem for the secondorder parabolic equations

Tahir S. Gadjiev*, Shafiga N. Mammadova

Department of Non-linear Analysis, Institute of Mathematics and Mechanics, 9,F.Agayev str., AZ1141, Baku, [email protected]; shafa [email protected]

2000 Mathematics Subject Classification. Primary 35J15; Secondary 35K10

Let Ω ⊂ Rn, n ≥ 2 be an arbitrary unbounded domain, x = (x1, ..., xn)bea point of this space. In the cylindrical domain D = Ω × (t > 0) considerthe first mixed problem for the second order linear parabolic equation

∂u

∂t=

n∑i,j=1

∂

∂xi

(aij(x, t)uxj

)+

n∑i=1

bi(x, t)∂u

∂xi+ c(x, t)u (1)

u|∂Ω = 0 (2)

u|t=0 = ϕ(x) (3)

In the given paper the questions on stabilization of solutions of the firstmixed problem for linear divergent parabolic equations with the lowest coef-ficients are considered. Conditions that connect the coefficients of equationwith the geometry of the domain, providing the estimation of solutions arefound. In this paper we indicated the class of domains, for which the esti-mation dependent on the geometry of domain is established. The lower andupper estimations of the solutions, which show an exact order of growth ofthe solution, are obtained. We proved the Harnack inequality under the def-inite conditions on the coefficients, connected with the geometry of domain.The Cauchy problem is also considered.

In the given paper the new method for linear divergent equations ispractically suggested. These results are extended on nonlinear divergentparabolic equations with the lowest terms.

The questions of stabilization for linear and nonlinear divergent equa-tions were studied by A. K. Gushin [1], F. Kh. Mukminov, I. M. Bikkulov[2], A. F. Tedeev [3], T. S. Gadjiev, Sh. N. Mammadova [4], S.Kamin [5].

References

[1] Gushin, A.K., On the behaviour as t → ∞of solutions of the second mixedproblem for a second order parabolic equation. Appl. Math. Optim. 6 (1980),169-180.



[2] Mukminov, F.Kh., Bikkulov, I.M., On stabilization of norm of solutions ofone mixed problem for the fourth and sixth order parabolic equations in theunbounded domain. (Russian). Math. Sbornik 195, (2004), no.4, 115-142.

[3] Tedeev, A.F., Stabilization of solutions of the first mixed problem for high orderquasilinear parabolic equations. (Russian). Differ. Uravn. 25 (1989), no.3, 491-498.

[4] Gadjiev, T.S., Mammadova, Sh.N., Stabilization of solutions of the first mixedproblem for divergent parabolic equations. (Russian). Dokl. Nats. Akad. NaukAzerbaijana, LXI (2005), no.4, 6-15.

[5] Kamin, S., On stabilization of solutions of the Cauchy problem for parabolicequations. Proc. Roy. Soc. Edinburgh, Sect. A, 76 (1976), no.1, 43-53.



A priori estimates for two-dimensional parabolic equations

A. Chaoui, A. Guezane-Lakoud*

Department of Mathematics, Faculty of Sciences, University Badji Mokhtar, B.P.12, 23000, Annaba. Algeria

2000 Mathematics Subject Classification. 35B45, 35D05, 35F15, 35K90

The motivation of this talk is a series of recent results obtained concern-ing one-dimensional Cauchy problems with variable domains of operationalcoefficients, we can mention the works [4, 5]. In this work we study a two-dimensional (in time) Goursat boundary value problem generated by a classof parabolic differential equations with operational coefficients possessingvariable domains. We assume that the operational coefficients are submit-ted to certain conditions in a Hilbert space H, then we show that thisproblem is well posed in Hadamard sense. The proofs are performed bygeneralization of the well-known method of energy inequalities, first we de-rive a priori estimates for strong solutions with the use of Yosida operatorapproximation, and then by using previous results, we show that the rangeof the operator generated by the posed problem is dense. More precisely,we solve the following problem: Lu =

∂u

∂t1+∂u

∂t2+A(t)u = f(t)

l1u(t1, t2) = u(t1, 0) = ϕ(t1); l2u(t1, t2) = u(0, t2) = ψ(t2),

where t = (t1, t2) ∈ D which is a bounded rectangle in R2, f , ϕ and ψ aregiven.

In the case where the operational coefficients have constant domains,various important results were proved under different assumptions, see [1,2]. In all these works, the proofs are obtained via a priori estimates, whichfollow from the energy inequalities method. Our results extend some onesobtained in [3, 4]. A similar technique as in [3-5] will be applied in thisstudy; however, the results aim at different type of equations and arisefrom distinct hypotheses.

References

[1] N. I. Brich & N. I. Yurchuk, Goursat’s problem for abstract second order lineardifferential equations. Diff. Uravn. 7, 1017-1030, (1971).

[2] A. Guezane-Lakoud & N. I. Yurchuk, Probleme aux limites pour une equationdifferentielle operationnelle de second ordre, Maghreb. Mathematical Review.V.6. N1, 39-48, (1997).



[3] A. Guezane-Lakoud, Functional differential equations with non-local boundaryconditions. E. J. D. E. Vol 2005, N 88, 1-8, (2005).

[4] F. E. Lomovtsev, A boundary value problem for even order differential equationswhose operator coefficients have variable domains. Diff.equations. 8, 1310-1322,(1994).

[5] F. E. Lomovtsev, Complete parabolic operator differential equations with vari-able domains of operator coefficients. Dok. Nats. Akad. Nauk. Belarusi. Vol 46N3, 55-59, (2001).



Elliptic problems with nonlocal boundary-value conditions

Pavel Gurevich

Department of Differential Equations and Mathematical Physics, People’sFriendship University of Russia, Ordjonikidze str. 3, Moscow, 117923, [email protected]

2000 Mathematics Subject Classification. 35J

We discuss various settings of elliptic problems with nonlocal boundary-value conditions of the following type:

P(x,Dx)u = f0(x), x ∈ Q,Bj(x,Dx)u+ Nju = fj(x), x ∈ ∂Q, j = 1, . . . ,m,

where Q ⊂ Rn is a bounded domain, P(x,Dx),B1(x,Dx), . . . ,Bm(x,Dx)corresponds to “local” elliptic boundary-value problem (with P(x,Dx) be-ing an elliptic operator of order 2m), and Nj are nonlocal operators. Theboundary of Q may contain angular points or edges. The nonlocal operatorsNj need not be small or compact perturbations.

T. Carleman (1932), M. Vishik (1952), F. Browder (1964), A. V. Bit-sadze and A. A. Samarskii (1969) were the first to study nonlocal ellipticproblems. Significant progress in the general theory of nonlocal problemshas been made by A. L. Skubachevskii (since the 1980s).

In this communication, a special attention is paid to the most com-plicated situation where the support of nonlocal terms can intersect theboundary ∂G. The main difficulties here are due to the fact that general-ized solutions of such a problem can have power-low singularities near somepoints of the boundary ∂G even if the boundary and the right-hand side(f0, f1, . . . , fm) are infinitely smooth [1, 2].

We formulate theorems on the (Fredholm) solvability of nonlocal prob-lems and give necessary and sufficient conditions under which the regularityof generalized solutions does not fail [2].

Applications to the plasma theory, control theory, biophysics, and theoryof multidimensional diffusion processes are mentioned.

Supported by Russian Foundation for Basic Research (project no. 05-01-00422)and by the Russian President Grant (project no. MK-980.2005.1).

References

[1] Skubachevskii, A. L., Elliptic Problems with Nonlocal Conditions Near theBoundary, Mat. Sb. 129 (171) (1986), 279–302; English transl.: Math. USSRSb. 57 (1987).



[2] Gurevich, P., The Consistency Conditions and the Smoothness of GeneralizedSolutions of Nonlocal Elliptic Problems, Advances in Differential Equations.To be published.



On the vibrations of a thin plate with a concentrated mass

D. Gomez* 1, M. Lobo1, E. Perez2

1Departamento de Matematicas, Estadıstica y Computacion; 21Departamento deMatematica Aplicada y Ciencias de la Computacion; Universidad de Cantabria,Av. de los Castros s/n. 39005 Santander. [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 35P, 74K

We look at the vibrations of an elastic, homogeneous and isotropic platethat contains a small region whose size depends on a small parameter ε. Thedensity is of order O(ε−m) in the small region, the so-called concentratedmass, and it is of order O(1) outside; m is a positive parameter. We assumethat the thickness plate h0 also depends on ε; indeed, we perform the studyin the case where h = εr with h2 = h2

0/12 and r ≥ 1.We consider the associated spectral problem in the framework of the

Reissner–Mindlin plate model, which takes into account the thickness plate.We address the asymptotic behavior of the eigenvalues ζε and the eigen-functions uε of this spectral problem when the parameter ε tends to zero.Depending on the values of m and r, there appear different limit behaviorsfor the eigenelements that we describe. We use methods of spectral per-turbation theory along with techniques of matched asymptotic expansions,which allow us to provide additional information on the structure of theeigenfunctions associated with the low frequencies (see [1]–[3]).

References

[1] Gomez, D., Lobo, M., Perez, E., On a vibrating plate with a concentrated mass,C. R. Acad. Sci. Paris 328 Serie IIb (2000), 495–500.

[2] Gomez, D., Lobo, M., Perez, E., On the vibrations of a plate with concentratedmass and very small thickness, Math. Meth. Appl. Sci. 26 (2003), 27–65.

[3] Gomez, D., Lobo, M., Perez, E., On the structure of the eigenfunctions for avibrating plate with a concentrated mass and very small thickness, In IntegralMethods in Science and Engineering. Theoretical and Practical Aspects (ed.byC. Constanda, Z. Nashed, D. Rollins). Birhauser, Boston 2006, 47–59.



Exact solutions for the generalized shallow water wave equationby the general projective riccati equation method

Cesar A. Gomez S.∗, Alvaro Salas+

∗Department of mathematics, Universidad Nacional de Colombia, Bogota ,[email protected]

+Department of mathematics, Universidad de Caldas, Manizales , [email protected]

2000 Mathematics Subject Classification. 35C05

Conte [1] presented a general ansatz to seek more new solitary wave solu-tions of some NLPDEs. More recently, Yan [4] developed Contes methodand presented the general projective Riccati equations method . In this pa-per we will use this method to construct more exact solutions (soliton andperiodic) for the generalized shallow water wave and the combined cosh-sinhGordon equation . We present a programm in mathematica and applied themethod to search solutions of NLPDEs.

Key words: Nonlinear differential equation; Travelling Wave Solution; Mathemat-ica; Projective Riccati Equation Method.

References

[1] Conte-MusetteR. Conte & M. Musette , Link betwen solitary waves andprojective Riccati equations, J. Phys. A Math. 25 (1992), 5609–5623.

[2] Inc-MahmutE. Inc & M. Ergt, New Exact Tavelling Wave Solutions forCompound KdV-Burgers Equation in Mathematical Physics, Applied Mathe-matics E-Notes, 2 (2002), 45–50.

[3] Mei-Zhang-JiangJ. Mei, H. Zhang & D. Jiang, New exact solutions fora Reaction-Diffusion equation and a Quasi-Camassa-Holm Equation, AppliesMathematics E-Notes, 4 (2004), 85–91.

[4] YanZ. Yan, The Riccati equation with variable coefficients expansion algo-rithm to find more exact solutions of nonlinear differential equation, MMRC,AMSS, Academis Sinica, Beijing 22 (2003), 275–284.

[5] YanZ. Yan, An improved algebra method and its applications in nonlinearwave euations, MMRC, AMSS, Academis Sinica, Beijing 22 (2003), 264–274.



[6] HeremanW.Hereman, Symbolic computation of exact solutions expressiblein hyperbolic and elliptic functions for nonlinear partial differential anddiffential-difference equations, Journal of Symbolic Computation.

[7] Wazwaz A.M. Wazwaz, The variable separated ODE and the tanh methodsfor solving the combined and the double combined sinh-cosh-Gordon equations,Applied Matematics and Computation (2005).



On the stability of the minimal solution to quasilineardegenerate elliptic equation

T. Horiuch*, P. Kumulin

Department of Mathematical Science,Ibaraki University, Mito, Ibaraki, 310, [email protected]; [email protected]


We shall investigate the stability of the minimal solutions of quasilinearelliptic equation with Dirichelet boundary condition given by

Lp(u) = λf(u), in Ω,u = 0, on ∂Ω,

where λ is a nonnegative parameter, Ω is a bounded domain of RN andLp(·)(p > 1) is the p-Laplace operator defined by Lp(·) = −div

(|∇·|p−2∇·)

).

We assume that f(t) is increasing on [0,∞) and strictly convex with f(0) >0. Here the minimal solution uλ is defined as the smallest solution amongall possible classical solutions. Then we shall establish the stability of theminimal solutions uλ for any λ > 0, that is to say, the nonnegativity of thefirst eigenvalue the linearized operator L′p(uλ)(·) − λf ′(uλ) on L2(Ω) willbe shown. In the lecture we shall give a similar result on the Bi-p-Laplaceoperator Mp(u) = ∆(|∆u|p−2∆u) as well.Theorem (P-Harmonis case) Let uλ be the minimal solution for λ ∈(0,+∞). Then the first eigenvalue of L′p(uλ) − λf ′(uλ) is nonnegative. Inother words, we have the Hardy type inequality:∫

Ω|∇uλ|p−2

(|∇ϕ|2 + (p− 2)

(∇uλ,∇ϕ)2

|∇uλ|2)dx ≥ λ

∫Ωf ′(uλ)ϕ2 dx, (1)

for any ϕ ∈ Vλ,p(Ω). Here Vλ,p(Ω) is defined by

Vλ,p(Ω) = ϕ ∈M(Ω) : ||ϕ||Vλ,p< +∞, ϕ = 0 on ∂Ω, (2)

||ϕ||2Vλ,p=∫Ω|∇uλ(x)|p−2|∇ϕ|2 dx (3)

and by M(Ω) we denote the set of all measurable functions on Ω.

References

[1] T. Horiuchi and P. Kumlin, On the minimal solution for quasilinear degenerateelliptic equation and its blow-up. Journal of Mathematics Kyoto University 44No.2 (2004) 381–439



Limit relative category applied to the critical points result forthe nonlinear Hamiltonian system

Q-Heung Choi, Tacksun Jung*

Department of Mathematics Education, Inha University, Incheon 402-751, Korea;Department of Mathematics, Kunsan National University, Kunsan 573-360,Koreaqheung@@inha.ac.kr; tsjung@@kunsan.ac.kr


LetH(z(t)) be a C1 function defined onR2n. Let z = (p, q), p = (z1, · · · , zn),q = (zn+1, · · · , z2n). In this paper we investigate the multiplicity of 2π−periodicsolutions of the following Hamiltonian system

p = −Hq(p, q), q = Hp(p, q).

The system can be written in a compact version

z = J(Hz(z)), (1.1)

where z : R → R2n, z = dzdt , J =

(0 −II 0

), I is the identity matrix on

Rn, H : R2n → R, and Hz is the gradient of H. Let E = W12,2([0, 2π], R2n).

We look for the weak solutions z = (p, q) ∈ E of (1.1); that is, z = (p, q)satisfies ∫ 2π

0[(p+Hq(z)) · ψ − (q −Hp(z)) · φ]dt = 0

for all w = (φ, ψ) ∈ E. Let a · b and | · | denote the usual inner product andnorm on R2n. Assume that H satisfies the following conditions:(H1) H ∈ C1(R2n, R), H(z) = o(|z|2) as |z| → 0.(H2) There exist 1 < p1 ≤ p2 < 2p1 + 1, αi > 0, βi ≥ 0 for i = 1, 2 suchthat

α1|z|p1+1 − β1 ≤ H(z) ≤ α2|z|p2+1 + β2 for every z ∈ R2n.

(H3) There exist 0 < q2

2 < q1 ≤ q2 < 2 and αi, τi > 0, βi ≥ 0 for i = 1, 2such that

α1eτ1|z|q1 − β1 ≤ H(z) ≤ α2e

τ2|z|q2 + β2 for every z ∈ R2n.

Our main results are as follows.



Theorem 1.1. Assume that H satisfies the conditions (H1), (H2). Thenthe Hamiltonian system (1.1) has at least two nontrivial 2π-periodic solu-tions.

Theorem 1.2. Assume that H satisfies the conditions (H1), (H3). Thenthe Hamiltonian system (1.1) has at least two nontrivial 2π-periodic solu-tions.



Representation Green function of the Dirichlet problems for thebi- harmonic equation

T. Sh. Kalmenov*, B. D. Koshanov

The Centre for Physical and Mathematical Researches, Ministry of Education andSciences of the Republic of Kazakhstan, Almaty, Shevchenko st. 28, Republic [email protected], Almaty, Shevchenko st. 28, Republic ofKazakhstan

2000 Mathematics Subject Classification. 35J67, 31A30, 31A10

In the theory of elasticity explicit representations of solutions of the Dirich-let problems for bi-harmonic of the equation plays an important role. Letus consider the following problem.

Find in an area Ωδ = x : ‖x − x0‖ < δ ⊂ Rn, n = 2m + 1 a functionu(x), satisfying to the following equation and boundary conditions:

∆2xu(x) = f(x), u|∂Ωδ

= 0,∂

∂nxu|∂Ωδ

= 0,

where ∂Ωδ = Sδ = x : ‖x‖ = δ.H. Begehr et others [1] for the first time explicitly solved a certain

Dirichlet problem for the inhomogenous polyharmonic equation in the unitdisc of the complex plane. But the problem of representation of Greenfunction in explicit form was remained unsolved.

The fundamental solution of the equation has a form [2] and be usingthe property of symmetry [3]:

|x− y

|y|2δ2|4−n|y

δ|4−n = |y − x

|x|2δ2|4−n|x

δ|4−n

we proved the following theorem:

Theorem. The Green function of the Dirichlet problem can be representedas:

Gn,2(x, y) =1

(4− n)4(2− n)· 1

(2π)n[|x− y|4−n − |x− y

|y|2δ2|4−n|y

δ|4−n]

+1

2 · 4 · (2− n)· 1

(2π)n· δ2[1− |y

δ|2][1− |x

δ|2]|x− y

|y|2δ2|2−n|y

δ|2−n.



References

[1] Begehr, H., Vanegas, C.J., Iterated Neumann problem for the higher order Pois-son equation. Math. Nachr. 279 (2006), p.38-57.

[2] Bers, L., Yonh, F., Schechter,M., Partial differential equations, Mir., Moscow,1966.

[3] Bitsadze, A. V., Equations of the mathematical physics, Nauka.,Moscow, 1985.



Some existence result to quasilinear elliptic equations

Tsang-Hai Kuo

Center of general education, Chang-gung University, 259, wen-hua 1st Rd.Kwei-shan, Tao-yuan, [email protected]

2000 Mathematics Subject Classification. 35D05

Consider the quasilinear elliptic equation−∑N

i,j=1 aij(x, u) ∂2u∂xi∂xj

+f(x, u,∇u) =0 on a bounded smooth domain Ω in RN with |f(x, r, ξ)| ≤ h(|r|)(1 +|ξ|θ), 0 ≤ θ < 2. We note that if there exist a super-solution ψ and a sub-solution φ with ψ, φ ∈ W 1,∞(Ω) and ψ ≥ φ a.e., then the existence ofsolutions is irrelevent to aij(x, r) for r beyond I0 = [−‖ψ‖∞, ‖φ‖∞]. It isshown that if the oscillation aij(x, r) with respect to r are sufficiently smallfor r ∈ I0 then there exists a solution u ∈W 2,p(Ω) ∩W 1,p

0 (Ω), 1 ≤ p <∞.

References

[1] L. Boccardo, F. Murat and J. P. Puel, Resultats d’existence pour certains prob-lemes elliptiques quasi-lineaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(1984),pp.213-235.

[2] L. Boccardo, F. Murat and J. P. Puel, Existence de solution faibles pour desequations elliptiques quasi-lineaires a croissance quadratique, Nonlinear PDEand their applications, College de France (Lions and Brezis editor)IV, ResearchNotes in Math. 84, Pitman, London(1983), pp19-73.

[3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Sec-ond Order , section edition, Springer-Verlag, New York, 1983.

[4] T. H. Kuo, Estimates on the approximation of solutions to certain quasilinearelliptic equations, Nonlinear Analysis, Vol.63, Issues 5-7, (2005), pp.e427-e434.

[5] T. H. Kuo and Y. J. Chen, Existence of strong solutions to some quasilinear el-liptic problems on bounded smooth domains, Taiwanese J. Math.6, No.2 (2002),187-204.



Kawahara equation in bounded domains

N. A. Larkin

Departamento de Matematica, Universidade Estadual de Maringa, Av. MarioClapier Urbinatti 182, zona 07, Maringa, Pr., 87020260. [email protected]


Initial boundary value problems in Q = (0, T ) × (0, 1) for the Kawaharaequation

ut + uux + aD3xu− bD5

xu = 0, (1)

where a, b are constant coefficients, are considered. When b = 0, a = 1 wehave the KdV equation. The Kawahara equation describes the dynamics oflong waves in a viscous fluid [1] and was considered in x ∈ R by variousauthors. On the other hand, mixed problems in bounded domains were notstudied. However, they have physical sense and are interesting as mathe-matical objects since boundary conditions for this equation of odd orderare not symmetric and a type of a corresponding mixed problem dependson a sign of the coefficient b.

First we consider in (0, 1) boundary value problems for the stationaryKawahara equation

cut + uux + aD3xu− bD5

xu = f(x),

where c is a positive constant, b may be positive or negative, which definesa corresponding boundary problem, and ab < 0. We prove the existence,uniqueness and continuous dependence of regular solutions on f(x). Thenusing discretization with respect to t and discrete Gronwall lemma, we provethe existence of local regular solutions to the mixed problem for (1). Globalregular estimates show that these local solutions may be prolonged for anyfinite T > 0. Finally, the exponential decay of L2-norms of solutions as ttends to infinity is proved.

References

[1] Topper J., Kawahara T., Approximate equations for long nonlinear waves in aviscous fluid. J. Phys. Soc. Japan 44 (1978), 663-666.



The resonant nonlinear Schrodinger equation and thereaction-diffusion system and the applications

Jyh-Hao Lee

Institute of Mathematics, Academia Sinica, Taipei, ROC (Taiwan)[email protected]

2000 Mathematics Subject Classification. 35Q51, 35Q55

I will make a brief report on the resonant nonlinear Schrodinger equationand the Reaction-Diffusion system and the application[1], [2]. A novel inte-grable version of the NLS equation namely ,

i∂ψ

∂t+∂2ψ

∂x2+

Λ4|ψ|2ψ = s

1|ψ|

∂2|ψ|∂x2

ψ. (1)

has been termed the resonant nonlinear Schrodinger equation (RNLS)[1].It can be regarded as a third version of the NLS, intermediate betweenthe defocusing and focusing cases. The additional nonlinear term in theNLS can be viewed as due to an additional electrostriction pressure ordiffraction term [4]. Even though the RNLS is integrable for arbitrary valuesof the coefficient s, the critical value s = 1 separates two distinct regions ofbehaviour. Thus, for s < 1 the model is reducible to the conventional NLS,(focusing for Λ > 0 and defocusing for Λ < 0). However, for s > 1 it is notreducible to the usual NLS, but rather to a Reaction-Diffusion system. Inthis case, the model exhibits novel solitonic phenomena [1].

The RNLS can be interpreted as an NLS-type equation with an addi-tional ‘quantum potential’ UQ = |ψ|xx/|ψ|. It is noted that the RNLS equa-tion, like the conventional NLS equation, may also be derived in the contextof capillarity models [2]. The RNLS equation is also related to an equationin plasma physics. It describes the propagation of one-dimensional longmagnetoacoustic waves in a cold collisionless plasma subject to a transversemagnetic field[3]. A Hirota bilinear representation of the Reaction-Diffusionsystem with non-zero boundary condition is given.Here one dissipaton andtwo-dissipaton exact solutions are obtained by Hirota bilinear method.Someplots of solutions will be also presented here. The results presented are jointworks with O. K. Pashaev.

References

[1] Pashaev ,O. K. and Lee, J.-H., Resonance solitons as black holes in Madelungfluid. M od. Phys. Lett. A 17 1601–1619 (2002).



[2] Rogers, C. and Schief, W. K., The resonant nonlinear Schrodinger equation viaan integrable capillarity model. I l Nuovo. Cimento 114, 1409–1412 (1999).

[3] Lee,J.-H.,Pashaev,O. K.,Rogers, C. and Schief, W. K.,The Resonant Nonlin-ear Schrodinger Equation in Cold Plasma Physics.Application of Backlund-Darboux Transformations and Superposition Principles.preprint 2006

[4] Hasegawa ,A., Kodama ,Y., Solitons in Optical Communications. ClarendonPress, Oxford, (1995).



Quasineutral limits of the Vlasov-Maxwell-Fokker-Planck system

Fucai Li*, Hsiao Ling, Shu Wang

Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China;Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing 100080, P.R. China; College of Applied Sciences, Beijing University ofTechnology, Pingleyuan 100, Beijing 100022, [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 35Q60, 82D10, 35B40

In the talk, we shall discuss the quasineutral limits of the Vlasov-Maxwell-Fokker-Planck (VMFP) system. It provides a statistic description of plasmain the terms of charged particle density f ε(t, x, ξ) depending on the timet ≥ 0, the position x = (x1, x2, x3) ∈ [0, 1]3 ≡ T3, and the velocity ξ =(ξ1, ξ2, ξ3) ∈ R3. The (rescaled) VMFP system takes the form

∂tfε + ξ · ∇xf

ε + (Eε + αξ ×Bε) · ∇ξfε = divξ

(βξf ε + σ∇ξf

ε),ε2α∂tE

ε − curlxBε = −α∫

R3ξf εdξ, −ε2divxE

ε = 1−∫

R3f εdξ,

α∂tBε + curlxEε = 0, divxB

ε = 0,

where ε, α, β, σ are positive parameters.Under an appropriate on initial data, we show that, as ε → 0, the

solution of the above VMFP system converges to the so-called electronmagnetohydrodynamics equations

∂tv + divx(v ⊗ v)− e− αv × b+ βv = 0,α∂tb+ curlxe = 0, divxv = 0αv − curlxb = 0, divxb = 0.

In fact, we obtain that the current J ε = ρεuε(t, x) =∫R3 ξf ε(t, x, ξ)dξ con-

verges weakly to v in L∞([0, T ], L1(T3)), the scaled electric field and themagnetic field converge strongly

εEε → 0 and Bε → b in L∞([0, T ], L2(T3))

respectively as ε, σ → 0.The details can be found in [3].

References

[1] Ling Hsiao, Fucai Li, Shu Wang, Coupled quasineutral and inviscid limit of theVlasov-Maxwell-Fokker-Planck system, preprint.



Global Entropy Solutions for a Nonstrictly Hyperbolic System

Yunguang Lu

Department of Mathematics, National University of Colombia, Cra. 30, Calle.45, Bogota, [email protected]

2000 Mathematics Subject Classification. 20K

An important class of the equations arising in applications are the nonlinearsystem of conservation laws. The basic question in this area is the existenceof solutions to these equations. This helps to answer the question if themodel of the natural phenomena at hand has been done correctly, if theproblem is well posed.

In the past years, we studied the existence of weak solutions for somenonlinear hyperbolic equations arisen in the compressible fluid and gas me-chanics. In this talk, we shall introduce an existence theorem for globalentropy solutions to the nonstrictly hyperbolic system

ρt + (ρu)x = 0ut + (u2

2 + P (ρ))x = 0,(1)

with bounded measurable initial data

(ρ(x, 0), u(x, 0)) = (ρ0(x), u0(x)), ρ0(x) ≥ 0, (2)

where the nonlinear function P (ρ) = θ2ρ

γ−1, θ = γ−12 and γ > 3 is a con-

stant.The method we use here is the theory of compensated compactness

coupled with some basic ideas of the kinetic formulation by Lions, Perthame,Souganidis and Tadmor[1, 2]

References

[1] P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of en-tropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerianand Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599–638.

[2] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropicgas dynamics and p-system, Commun. Math. Phys., 163 (1994), 415–431.

[3] Y.-G. Lu, Hyperboilc Conservation Laws and the Compensated CompactnessMethod, Vol. 128, Chapman and Hall, CRC Press, New York, 2003.

[4] Y.-G. Lu, Existence of Global Entropy Solutions to a Nonstrictly HyperbolicSystem, Arch. Rat. Mech. Anal., 178(2005), No. 2, 287–299.



Unified approach to solve inverse problems for partialdifferential equations

Sadia Makky*, Ali Sayfy1, Ahlam Jamil2

Owens College, USA, 1American University of Sharjah, UAE, 2University ofBaghdad, Iraqsadia [email protected]; [email protected]

2000 Mathematics Subject Classification. 65N21

The direct variational method is used successfully to solve a variety of in-verse problems. The types of the governing differential equations consideredcover a wide spectrum, including elliptic, parabolic, systems of equations,etcetera. In addition, the auxiliary conditions range from boundary condi-tions, where the boundary is fixed, to cases where the boundary is free ormoving. Incorporating multi-objective optimization in connection with thevariational approach, inverse problems with auxiliary conditions specifiedat randomly chosen points of the region can be tackled. The essence ofthe method is to: i) Find a functional whose critical point is a solution forthe differential equation. ii) Choose the feasible functions for the variationalproblem to be functions satisfying the given boundary, initial, and auxiliaryconditions. iii) Incorporate the unknown boundary (in case of free or mov-ing boundary condition) as one of the unknown functions in the variationalproblem.

References

[1] Mahlol, M., Inverse problems in differential equations with an application tolocalizing brain tumors, Ph.D. thesis, University of Baghdad., (1993).

[2] Makky S. M., Ali, J., Direct and inverse problems of acoustic wave scattering,Mu’tah Journal for Research and Studies, Natural and Applied Science Series,11, (1996).

[3] Sayfy, A., Makky S., Numerical solution of inverse problem for elliptic partialdifferential equations, Intern. J. Computer Math., 80(5), (2003), 665-670.

[4] Somersalo, E., Chenny, M., Issacson, D., Issacson, E., Layer stripping : a directnumerical method for impedance imaging, Inverse Problems, 7, (1991), no. 6,899-926.



Fourth order elliptic equation with critical exponent oncomplete manifolds

Youssef Maliki

Department of mathematics ,University Aboubekr Balkaid of Tlemcen, Faculty ofSciences BP 119, [email protected]


Let (M, g) be an n−dimensional complete noncompact Riemannian man-ifolds with n ≥ 5. We use the variational method to investigate existenceof solutions u ∈ C4,α

loc (M) , α ∈ (0, 1) to the fourth order nonlinear ellipticequation

∆2gu+ div (a(x)∇gu) + b(x) = f(x) |u|N−2 u (1)

where ∆gu = −div (∇gu)is the Laplace-Beltrami operator, a, b, f are smoothfunctions onM andN = 2n

n−4 is the critical Sobolev exponent. This equationhas geometrical roots, it arises from the study of the so-called Q−curvaturewich can be interpreted by mean of the equation

Pgu =n− 4

2Qu

n+4n−4

where

Pg = ∆2gu+ div(anSg + bnRic)du+

n− 42

Q

is the well known Panietz-Branson operator, and

Q = cn |Ric|2 + dnS2g −

12 (n− 2)

∆gSg.

Sg, Ric denote respectively the scalar and Ricci curvature, and an, bn, dn

are dimensional constants.Equation(1) is investigated in the case of compact manifold by D.Caraffa

[2] firstly, for f constant and secondly, for f a positive function.The authorused the variational approach introduced by Yamabe when he attemptedto solve his equation.

On complete noncompact Riemannian manifold , conditions on the ge-ometry of the manifold together with conditions on the decay of the functionf at infinity must be added.

We present the main result as follows:



Theorem 1. Let M be a complete noncompact riemannian n−manifold(n > 6) with bounded geometry, in the sens that the Ricci curvature isbounded from below and the injectivity radius is positive. Let a, b,and fbe smooth real valued functions on M.Suppose that the operator I (u) =∫M |∆gu|2 dvg −

∫M a(x) |∇gu|2 dvg +

∫M b(x)u2 is coercive . Under the fol-

lowing assumptions: (1) The functions a, b, f are bounded , f is nonnegativeand ∫

Mf (x) dvg <∞,

∫Ma (x) dvg <∞,

∫Mb (x) dvg <∞,

(2) There exists a positive constant C such that

|∆gf | < Cf, and

∫MfN+1dvg <∞,

(3)For any ε > 0 there exists a compact set K such that f ≤ ε on M −K,and (4)At a point xo where f is maximal we have

scal(xo) +2n (n− 1)n2 − 2n− 4

a(xo)− (n− 4) (n− 6) (n+ 2)4 (n2 − 2n− 4)

∆gf (xo)f (x0)

> 0,

There exists a solution u ∈ H22 (M) of (1) . Moreover, u ∈ C4,β

loc (M) ,for some β ∈ (0, 1) .

.

References

[1] T. Aubin, Some nonlinear problems in Riemannian geometry, Springer mono-graphs in mathematic. Springer Verlage,(Brelin)1998.

[2] D.Craffa,,Equations elliptiques du catrieme ordre avec exposent critiques surles varietes Riemanniennes compactes, J.Math.Pures et Appl.,80,9(2001),941-960.

[3] Hebbey, Sbolev spaces on Riemannian manifolds, Lecture Notes in Mathemat-ics, Berlin, vol.1635,1996.

[4] D.E.Edmunds, D.Fourtunato and E.Janelli, Critical exponents, critical dimen-sions, and the biharmonic operator, Arch Rational Mech.Anal., 112(1990), 269-289.

[5] The Yamabe problem,Bulletin of the American Mathematical Society (N.S.),17(1987),no. 1, 37-91.

[6] P. L. Lion, The concentration-compactness Principle in the calculs of variations.The limite case, Part1, Rev.Mat. Iberoamericana, 1(1985), 145-201.



Smooth null-solutions to PDEs with several Fuchsian variables

Takeshi Mandai

Research Center for Physics and Mathematics, Osaka Electro-CommunicationUniversity, 18-8 Hatsu-cho, Neyagawa-shi, Osaka 572-8530, [email protected]

2000 Mathematics Subject Classification. 35A07 (35L80)

N. S. Madi([4]) extended the notion of Fuchsian partial differential equationsdefined byM. S. Baouendi and C. Goulaouic([1]) to those with several Fuchsian vari-ables. For such an operator P = P (x, y;Dx, Dy) with (x, y) = (x1, . . . , xp, y1, . . . , yq)where x are Fuchsian variables, a distribution u near (x, y) = (0, 0) is calleda null-solution, if Pu = 0 and (0, 0) ∈ suppu ⊂ (x, y) | xj ≥ 0(∀j).The author and his collaborators([2]) showed the existence of distributionnull-solutions under some assumptions, which is also an extention of theresult for the case with one Fuchsian variable([3],[5]). There is, however,a big difference between the case with one Fuchsian variable where thereare no C∞ null-solutions, and the case with more Fuchsian variables whichinclude some cases where there can be a C∞ null-solution. This talk givessome examples and results to make this difference clearer.

One of the main results is about the typical case, where there are onlytwo variables which are both Fuchsian. Let P =

∑i,j≤m ai,j(x1, x2)Di

x1Dj

x2

be a partial differential operator with respect to x = (x1, x2) and as-sume that P is Fuchsian with weight 0 with respect to (x1, x2) in thesense of Madi, i.e. ai,j(x1, x2) = xi

1xj2ai,j(x1, x2) where ai,j are real-analytic

near (0, 0). We assume that the principal symbol has the form pm(x; ξ) =∏ml=1(x1ξ1− λl(x)x2ξ2) where λl are distinct real-valued real-analytic func-

tions. Then, we have the following.

1. If there exists l such that λl(0, 0) > 0, then there exists a C∞ null-solution u.

2. If λl(0, 0) < 0 for every l, then there exists no C∞ null-solutions, butthere exists a distribution null-solution.

References

[1] Baouendi, M. S. & Goulaouic, C., Cauchy problem with characteristic initialhypersurface, Comm. Pure Appl. Math. 26 (1973), 455–475.



[2] Belarbi, M., Mandai, T. & Mechab, M., Null solutions for partial differentialoperators with several Fuchsian variables, Math. Nachr. 273 (2004), 1–11.

[3] Igari, K., Non unicite dans le probleme de Cauchy caracteristique — cas detype de Fuchs, J. Math. Kyoto Univ. 25(1985), 341–355.

[4] Madi, N. S., Solutions locales pour des operateurs holomorphes fuchsiens enplusieurs variables, Ann. Math. Pura Appl. (IV) 163(1993), 1–15.

[5] Mandai, T., Existence of distribution null-solutions for every Fuchsian partialdifferential operator, J. Math. Sci. Univ. Tokyo 5(1998), 1–18.



Quasilinear equations with quadratic growth in ∇w and largesolutions for semilinear equations

David Arcoya, Pedro J. Martınez-Aparicio*

Dpto. Analisis Matematico, Univ. Granada, 18071 Granada, [email protected]

In this work we prove the existence of positive solution for the problem

−∆w + h(w)|∇w|2 = k(x) en Ωw ∈ H1

0 (Ω)

where Ω ⊂ RN (N ≥ 2) is a bounded and smooth domain, k ∈ L∞(Ω),k(x) > 0 and h : (0,+∞) −→ [0,+∞) is a function that satisfies

lims→0+

sh(s) < +∞.

We improve results in [2] to handle singular functions h at s = 0.We will also present an application to the existence of solutions to theboundary value problem having blow-up at the boundary. Specifically, wesolve the b.v.p.

∆u = k(x)f(u), en Ω

limdist(x,∂Ω)→0

u(x) = +∞.

Contrary to the previous works [1, 3, 4] we don’t assume f to be nondecrea-sing.

References

[1] C. Bandle and M. Marcus, Large solutions of semilinear Elliptic Equations:Uniqueness and Asymptotic Behaviour, Journal D’Analyse Mathematique, Vol.58 (1992), 9-24.

[2] L. Boccardo, F. Murat y J.P. Puel, Existence de solutions faibles pour desequations elliptiques quasi-lineaires a croissance quadratique, Nonlinear partialdifferential equations and their applications. College de France Seminar, Vol.IV (Paris, 1981/1982), 19–73, Res. Notes in Math., 84, Pitman, Boston, Mass.-London, 1983.

[3] J. Garcıa Melian, Boundary behaviour for large solutions to elliptic equationswith singular weights, Sometido.

[4] A.V. Lair, A Necessary and Sufficient Condition for Existence of Large Solu-tions to Semilinear Elliptic Equations, Journal of Mathematical Analysis andApplications, 240 (1999), 205-218.



Mathematical modelling of a nuclear waste disposal site viahomogenization

Alain Bourgeat, Olivier Gipouloux, Eduard Marusic-Paloka*

Universite Lyon 1, Lyon, France; Universite de St-Etienne, St-Etienne, France;Department of mathematics, University of Zagreb, 10 000 Zagreb, [email protected]

2000 Mathematics Subject Classification. 35B40, 35Q20, 35B27, 76Q05

The goal of our work is to give an accurate model describing the globalbehavior of an underground waste repository array, made of a high numberof units containing waste, once the units start to leak. The purpose of sucha global model is to be used for the full field three dimensional simulationsnecessary for safety assessements. The disposal site can be described as arepository array made of high number of units inside a low permeabilitylayer, called host layer, like for example clay, included between layers withhigher permeability (e.g. limestone). There is a flow crossing the repositoryarray which is produced by the pressure drop, or hydraulic head, in the re-gion. The pollutant is then transported both by the convection produced bythe water flowing slowly (creeping flow) through the rocks and by the dif-fusion coming from the dilution in the water. The general transport model,we start from, will also include possible chemical effects and radioactivedecay, following the test case [4]. We study the worst possible scenario inwhich all units start leaking at the same time, either due to some outerfactor or due to simple aging of materials used to build the containers. Theratio between the width of a single unit l and the layer length L, can beconsidered as a small parameter, ε, in the detailed microscopic model. Thestudy of the renormalized model behavior, as ε tends to 0, by means of thehomogenization method and boundary layers, gives an asymptotic modelwhich could be used as a global repository model for numerical simulations.Parts of our work were presented in papers [1],[2],[3].

References

[1] Bourgeat A., Gipouloux O., E. Marusic-Paloka E., Mathematical Modelling ofan array of underground waste containers. C.R.Acad.Sci.Paris, Mecanique 330,371-376, 2002.

[2] Bourgeat A., Gipouloux O.,Marusic-Paloka E., Modeling of an undergroundwaste disposal site by upscaling, Mathematical Models in Applied Sciences, 27(2004), 381-403.



[3] Bourgeat A., Marusic-Paloka E., A homogenized model of an undergroundwaste repository including a disturbed zone, Multi-Scale Modeling and Simu-lation - SIAM,Vol 3, No 4 (2005), 918-939.

[4] Exercice COUPLEX at https://mcs.univ-lyon1.fr/MOMAS



Attractors for 2D-Navier-Stokes equations with delays on someunbounded domains

Pedro Marın-Rubio*, Jose Real

Dpto. Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla,[email protected]; [email protected]

2000 Mathematics Subject Classification. 35B41, 35Q30, 35B40, 35R10

We consider the following Navier-Stokes problem with delay terms:

∂u

∂t− ν∆u+

2∑i=1

ui∂u

∂xi= f(t)−∇p+ g(t, ut) in (0, T )× Ω,

divu = 0 in (0, T )× Ω,u = 0 on (0, T )× Γ,u(0, x) = u0(x), x ∈ Ω,u(t, x) = φ(t, x), t ∈ (−h, 0) x ∈ Ω,

Here, the domain Ω ⊂ R2 is an open set that it is not necessarily boundedbut satisfies a Poincare inequality.

Under suitable assumptions on the terms f and g, existence and unique-ness of solution for this problem (and more general versions) were studiedin [3]. Observe that compactness method is not valid here directly.

In this talk we will discuss on the existence of two types of attractorsfor some associated processes, circumventing again the lack of compact em-bedding (oppositely to [2]).

Our proof uses an energy method [5, 1] and is valid for the autonomousand non-autonomous cases, and in the frameworks of tempered and non-tempered universes, solving positively the invariance question for both at-tractors.

References

[1] Caraballo, T., Lukaszewicz, G. & Real, J. Pullback attractors for asymptoti-cally compact non-autonomous dynamical systems, Nonlinear Analysis, TMA64 (2006), 484–498.

[2] Caraballo, T. & Real, J. Attractors for 2D-Navier-Stokes models with delays,J. Differential Equations 205 (2004), 271–297.

[3] Garrido-Atienza, M. J. & Marın-Rubio, P. Navier-Stokes equations with delayson unbounded domains, Nonlinear Analysis, TMA 64 (2006), 1100–1118.



[4] Moise, I., Rosa, R. & Wang, X. Attractors for noncompact nonautonomoussystems via energy equations, Discrete and Continuous Dynamical Systems 10(2004), 473–496.

[5] Rosa, R. The global attractor for the 2D Navier-Stokes flow on some unboundeddomains, Nonlinear Analysis, TMA 32 (1998), 71–85.



Lyapunov inequalities for differential equations with applicationsto nonlinear problems

A. Canada, J. A. Montero*, S. Villegas

Department of Mathematical Analysis , University of Granada, 18071 Granada,[email protected]

2000 Mathematics Subject Classification. 34B05, 34B15, 35J25, 35J65

This talk is devoted to the study of Lp Lyapunov-type inequalities (1 ≤ p ≤∞) for linear differential equations. The well-known Lyapunov inequalitystates that if a ∈ L∞(0, L), then a necessary condition for the b.v.p.

u′′(x) + a(x)u(x) = 0, x ∈ (0, L), u′(0) = u′(L) = 0 (1)

to have nontrivial solutions is:∫ L0 a+ dx > 4/L, where a+(x) = maxa(x), 0

(see for instance [3]). Previous condition is enunciated in terms of the L1-norm of the coefficients functions of the considered equation. In this talkwe provide new results on Lp Lyapunov-type inequalities with 1 < p ≤ +∞for ordinary differential equations. We show a qualitative and quantitativetreatment of the problem (see [1] ).

In the case of partial differential equations on a bounded and regulardomains in RN , we present a qualitative discussion and it is proved thatthe relation between the quantities p and N/2 plays a crucial role, showinga deep difference with respect to the ordinary case ([2]).

In the proof, the best constants are obtained by using variational meth-ods, including direct methods and Lagrange multiplier theorem.

The linear study is combined with Schauder fixed point theorem toprovide new conditions about the existence and uniqueness of solutions forresonant nonlinear problems. Related problems can be found in [4, 5].

References

[1] A. Canada, J.A. Montero and S. Villegas, Liapunov-type inequalities and Neu-mann boundary value problems at resonance, Mathematical Inequalities andApplications, 8 (2005), 459-476.

[2] A. Canada, J.A. Montero and S. Villegas, Lyapunov inequalities for partialdifferential equations, To appear in the Journal of Functional Analysis.

[3] P. Hartman, Ordinary Differential Equations, John Wiley and Sons Inc., NewYork-London-Sydney, 1964.



[4] W. Huaizhong and L. Yong, Neumann boundary value problems for second-order ordinary differential equations across resonance, SIAM J. Control andOptimization, 33, (1995), 1312-1325.

[5] J. Mawhin, J.R. Ward and M. Willem, Variational methods and semilinearelliptic equations, Arch. Rational Mech. Anal., 95, (1986), 269-277.



Transmission problems related to the interaction of metallic andpiezoelectric materials

David Natroshvili

Department of Mathematics, Georgian Technical University, 77 M.Kostava str.,Tbilisi 0175, [email protected]

2000 Mathematics Subject Classification. 35J55, 74F05, 74F15, 74B05

We study the following mathematical problem related to engineering appli-cations: Given is a three-dimensional composite consisting of a piezoelectric(ceramic) matrix with metallic inclusions (electrodes). We derive a linearmodel for the interaction of the corresponding 4-dimensional thermoelasticfield in the metallic part and 5-dimensional thermoelectroelastic field in thepiezo-ceramic part.

The main difficulty in the modelling is to find appropriate boundary andtransmission conditions for the composed body. The mathematical analysisincludes then the study of existence, uniqueness and regularity of the re-sulting elliptic boundary-transmission problem assuming the metallic andceramic materials occupy smooth or polyhedral domains.

With the help of the indirect boundary integral equations method wereduce the complex transmission problem to the equivalent strongly ellipticsystem of pseudodifferential equations involving pseudodifferential opera-tors on manifold with boundary. The solvability and regularity of solutionsto these boundary integral equations and the original transmission problemare analyzed in Sobolev-Slobodetski (Bessel potential) Hs

p and Besov Bsp,t

spaces. This enables us to investigate also stress singularities which appearnear zones, where the boundary conditions change and where the interfacemeets the exterior boundary. We show that the order of the singularityis related to the eigenvalues of the symbol matrices of the correspondingpseudodifferential operators and study their dependence on the materialconstants of the composite.



Analytical behavior of 2-D incompressible porous flow

Rafael Orive

Departamento de Matematicas, Universidad Autonoma de Madrid, Crta.Colmenar Viejo km. 15 28049 Madrid, [email protected]

2000 Mathematics Subject Classification. 76S05

The dynamics of a fluid through a porous medium is a complex and notthoroughly understood phenomenon. In [1] we study the analytic structureof a two-dimensional mass balance equation in porous medium (PM). Weobtain the existence and uniqueness of the solutions for the (PM) by theparticle-trajectory method. Also, we show some blow-up criterions and somefacts analogous for the two-dimensional quasi geostrophic equation thermalactive scalar (QG). Detailed mathematical criteria are developed as diag-nostics for self-consistent numerical calculations indicating nonformationof singularities. Finally, we obtain blow-up results in a class of solutionswith infinite energy of a two-dimensional mass balance equation in porousmedium.

References

[1] Cordoba, D., Gancedo, F., Orive, R., Analytical behavior of 2-D incompressibleflow in porous media, preprint submitted to J. Math. Phy. (2006).



On a Nonlinear Eigenvalues Problem

Isselkou Ould Ahmed Izid Bih

FST de Nouakchott, B.P. 5026 Nouakchott, [email protected]

2000 Mathematics Subject Classification. 35P30, 35J60

We consider the problem

(Pαλ )

∆u+ λ(1 + u)α = 0, in B1;u > 0, in B1;u = 0, on ∂B1

where B1 is the unit ball of <n, n ≥ 3, λ > 0 and α > 1.This problem arises in many physical models(cf. [1], [2]). It is well known(cf. [2],[3]) that there exists a constant λ∗(α), such that (Pα

λ ) admits, atleast, one solution if 0 < λ < λ∗(α) and no solution if λ > λ∗(α). As fareas the author is aware, very little is known about this constant and thecorresponding solutions. Let φ be the Lane-Emden function([1]) in the n-dimensional space.For every ρ > 0 and n ≥ 3, one defines, ψρ(r) = φ(ρr)−φ(ρ)

φ(ρ) , r ∈ [0, 1].We show that, If 1 < α and r0 is the first ”zero” of φ, then

λ∗(α) = maxr∈[0,r0[

r2φα−1(r).

If 1 < α < n+2n−2 and λ = λ∗(α), there exists a unique rλ∗(α) > 0, such

thatλ∗(α) =

(rλ∗(α)

)2φα−1(rλ∗(α)) and the unique solution of (Pα

λ∗(α)) is ψrλ∗(α).

When 0 < λ < λ∗(α), there exists two positive constants rλ and ρλ, suchthat rλ < rλ∗(α) < ρλ < r0, λ = r2λφ

α−1(rλ) = ρ2λφ

α−1(ρλ), the prob-lem (Pα

λ ) admits the two solutions, uλ = ψrλ, vλ = ψρλ

; limλ→0 uλ =0, in C0(B1) andlimλ→0 vλ(r) = ∞, ∀ r ∈ [0, 1[.

If α = n+2n−2 , then λ∗(α) = n(n−2)

4 , rλ∗(α) =√n(n− 2),

rλ =

√1− 2λ

n(n−2) −√

1− 4λn(n−2)

(n(n− 2))−1√

2λ, ρλ =

√1− 2λ

n(n−2) +√

1− 4λn(n−2)

(n(n− 2))−1√

2λ,

limλ→0 uλ = 0, limλ→0 vλ(r) = r2−n − 1, ∀ r ∈]0, 1].

If n+2n−2 < α and λs = 2

(α−1)2(α(n−2)−n), then φ(r) ∼ λ

1α−1s r

21−α , as r →

∞.



References

[1] CHANDRASEKHAR, S., An Introduction to the Study of Stellar Structure,inUniversity of Chicago Press, Chicago 1939.

[2] CRANDALL, M.G., RABINOWITZ, P.H., Some Continuation and VariationalMethods for Positive Solutions of Nonlinear Elliptic Eigenvalue Problems,Arch. Rational Mech. Anal. 58 (1975), 207-218.

[3] JOSEPH, D.D., LUNDGREN,T.S., Quasilinear Dirichlet Problems Driven byPositive Sources, Arch. Rational Mech. Anal. 49 (1973), 241-269.



Identification of degenerate relaxation kernels in a heat flowwith flux-type additional conditions

Enno Pais

Institute of Mathematics, Tallinn University of Technology, Ehitajate rd 5, 1Tallinn, [email protected]

2000 Mathematics Subject Classification. 35R30

An inverse problem to find degenerate time- and space-dependent relaxationkernels of internal energy and heat flux in one-dimensional heat flow isconsidered. The following heat equation is studied

β(x)∂

∂tu(x, t) +

∂

∂t

t∫0

n(x, t− τ)u(x, τ) dτ =∂

∂x(λ(x)ux(x, t))

− ∂

∂x

t∫0

m(x, t− τ)ux(x, τ) dτ + r(x, t), x ∈ (0, 1), t > 0.

The degenerate memory kernels have the forms

n(x, t) =N1∑j=1

νj(x)nj(t), m(x, t) =N2∑k=1

µk(x)mk(t) ,

where νj , µk are given x-dependent functions and nj , mk are unknowntime-dependent coefficients. Additional information to recover these kernelsconsists of a finite number N = N1 +N2 measurements of heat flux in fixedpoints over the time

q(xi, t) = −λ(xi)ux(xi, t) +t∫

0

m(xi, t− τ)ux(xi, τ) dτ = hi(t),

i = 1, . . . , N, t > 0 , where hi are given functions.Existence and uniqueness of a solution of the inverse problem are proved.Additional information to recover the memory kernels can consist of purelytemperature observations [1, 3] or simultaneous observations of temperatureand heat flux [1]. Paper [2] deal with the simplified case when the modelcontains only the relaxation kernel of heat flux.



References

[1] J. Janno and L. v. Wolfersdorf, Identification of memory kernels in generallinear heat flow, J. Inv. Ill-Posed Problems 6 (1998), 141–164.

[2] J. Janno and L. v. Wolfersdorf, Identification of a special class of memorykernels in one-dimensional heat flow, J. Inv. Ill-Posed Problems 9 (2001), 389–411.

[3] E. Pais and J. Janno, Identification of two degenerant time- and space-dependent kernels in a parabolic equation, EJDE 108 (2005), 1–20 (2005).



Bifurcations in a thermoconvective problem with variableviscosity

Henar Herrero, Olivier Lafitte and Ana Marıa Mancho, Francisco Pla*

Depto. de Matematicas, Facultad de Ciencias Quımicas, Universidad deCastilla-La Mancha 13071 Ciudad Real, [email protected]

2000 Mathematics Subject Classification. 76E06, 76M99

A Boussinesq Navier-Stokes problem with temperature-dependent viscosityin a plane parallel layer is presented. This problem has important applica-tions in mantle convection [1, 2].

The existence of a stationary bifurcation is proved together with a con-dition to obtain the critical parameters at which the bifurcation takes place[3]. With appropriate changes of variables and the decomposition into nor-mal modes we have simplified the expression to obtain a system of ordinarydifferential equations which is numerically manageable.

σΘ =(D2 − |k|2

)Θ +W

Rk2Θ = D(P − σ

Pr

)DW − k2

(P − σ

Pr

)W,

(1)

where Pf = D (ν (z) ·Df) − k2 (ν (z) · f), W the vertical velocity compo-nent, Θ the temperature, ν the variable viscosity and the boundary condi-tions are

W = DW = Θ = 0, on z = ±1/2. (2)

In reference [3] it is demonstrated that the problem (1) with the bc(2) has a unique solution (Θ,W ) ∈ H1

0 ×H20 . For exponential dependence

of viscosity with temperature [4, 5] the critical bifurcation curves and themost unstable modes have been numerically computed. The stability curvedemonstrates coherence in the results, so for different exponential rates thecritical wave number increases slightly with the exponential rate. Depen-dence on the exponential parameter is also presented. A higher values ofthe exponential rate in viscosity favors the instability.

References

[1] J. Frohlich, P. Laure, and R. Peyret, Large departures from Boussinesq approx-imation in the Rayleigh–Benard problem, Phys. Fluids, A 4 (1992), 1355–1372.



[2] M. Manga, D. Weeraratne and S.J.S. Morris, Boundary-layer thickness andinstabilities in Benard convection of a liquid with a temperature-dependent vis-cosity, Phys. Fluids, 13 (2001), 802-805.

[3] F. Pla, H. Herrero, O. Lafitte and A. M. Mancho, Bifurcations in a convec-tion problem with temperature-dependent viscosity, submitted to SIAM J. Appl.Math.

[4] L.-N. Moresi and V.S. Solomatov, Numerical investigation of 2D convectionwith extremely large viscosity variations, Phys. Fluids, 7 (1995), 2154–2162.

[5] Y. Ke and V.S. Solomatov, Plume formation in strongly temperature-dependentviscosity fluids over a very hot surface, Phys. Fluids, 16 (2004), 1059–1063.



Gaussian bounds for degenerate elliptic second order operatorson Rn and the weighted Kato square root problem

David Cruz-Uribe SFO, Cristian Rios∗

Department of Mathematics, Trinity College, 300 Summit Street, Hartford, CT,[email protected]

2000 Mathematics Subject Classification. 35R05, 35J70, 42B20, 47B44

The full Kato conjecture for elliptic operators in Rn was finally proved inthe sensational article [1]. If L = −div (A∇) is a uniformly complex ellipticoperator with bounded measurable coefficients in Rn, then the domain of thesquare root operator

√L is the Sobolev space H1(Rn) in any dimension with

the estimate ‖√Lf‖2 ∼ ‖∇f‖2. In this short communication we will present

some recent progress on the solution of the Kato square root problem foroperators with degenerate ellipticity. In particular, we consider operatorsof the form Lw = −w−1div (A∇), where w(x) is a nonnegative function inthe Muckenphoupt class A2(Rn), and the square matrix A(x) satisfies the(degenerate) complex ellipticity condition λw (x) |ξ|2 ≤ Re 〈Aξ, ξ〉 = Re

n∑i,j=1

Aij (x) ξj ξi,

| 〈Aξ, η〉 | ≤ Λw(x)|ξ · η|,

for ξ, η ∈ Cn and for some λ, Λ such that 0 < λ ≤ Λ < ∞. We prove thatif the matrix A is close to a real symmetric matrix (in the normalized L∞

norm) then the kernel of e−tLw satisfies standard Gaussian bounds in Rn+1+ .

We will also present some positive steps towards the solution of the Katosquare root problem for degenerate elliptic operators.

References

[1] Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan;Tchamitchian, Philippe, The solution of the Kato square root problem forsecond order elliptic operators on Rn, Ann. of Math. (2) 156 (2002), no. 2,633–654.



Resolution on n-order functional – differential equations withoperator coefficients and delay in Hilbert spaces

Chan RoathDepartment Scientific Research, Ministry of Education, Youth and Sports,#23,St.360, Sangkat Beung Keng Kang I, Khan Chamkamorn, Phnom Penh,[email protected]

2000 Mathematics Subject Classification. 34K30

We introduce the n-order functional-differential equation with operator co-efficients and delay in Hilbert spaces:

LnpoU(t) = f(t), Dn

t U(t)−n−1∑k=0

m∑j=0

[Akj +Akj(t)]Shkj+hkj

(t)Dkt U(t) = f(t)

(1)where Ln

po : Xn,α

Rt0+

−→ Y 0,α

Rt0+

, Dkt = dk

ikdtk, Rt0

+ = t ≥ t0 Xn,α

Rt0+

-Hilbert

space, containing functions with norm:

‖ U(t) ‖2=∫ ∞

t0exp(2αt)(

n−1∑k=0

‖ U (k)(t) ‖2x + ‖ U (n)(t) ‖2

y)dt, t0 ≥ −∞, α ∈ R

Y 0,α

Rt0+

-Hilbert space, containing functions with norm:

‖ U(t) ‖2=∫ ∞

t0exp(2αt) ‖ U(t) ‖2

y dt, t0 ≥ −∞, α ∈ R

Shkj+hkj(t)U(t) , U(t− hkj − hkj(t))- operator translation.

Necessary and sufficient conditions for equation (1) to have unique so-lution are:

Akj(t) ≡ 0, hkj(t) ≡ 0, k ≥ 0, j ≥ 0

In case the coefficients slightly variate, sufficiency is also satisfied.

References

[1] Agmon S., Nirenberg L. Properties of solutions of ordinary differential equa-tions in Banach space. Comm.appl.math., 1963, V.XVI, p.121-139.

[2] Kato T. On linear differential equations in Banach spaces. Comm. On Pureand Appl. Math., 1956, V.9, p.479-486.

[3] Minorsky. Self-excited oscillations in dynamical systems prossessing retard ac-tions. J.Appl.Mech.9 (1942), p67-71.



Iterative schemes for the existence of regular solution ofKorteweg’s model

F. Guillen-Gonzalez, M. A. Rodrıguez-Bellido*

Dpto. de Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla,Aptdo. 1160, 41080 Sevilla, [email protected]; [email protected]


We present a Navier-Stokes type model (in velocity) with a specific stresstensor introduced for the first time by Korteweg in 1901, that takes intoaccount that a nonuniform density (concentration or temperature) distri-bution induces stresses and convection in a fluid.

If the fluid is confined in a three-dimensional smooth enough domain Ω,the model can be written as:

(KM)

∂tρ+ (u · ∇)ρ = 0 in (0, T )× Ω,

∂tu− ν∆u + u · ∇u +∇p = −k∇ρ∆ρ in (0, T )× Ω,

∇ · u = 0 in (0, T )× Ω,

u = 0 on (0, T )× ∂Ω,

u|t=0 = u0, ρ|t=0 = ρ0 in Ω,

where u(t,x) is the solenoidal velocity, p(t,x) is the fluid pressure, ρ(t,x)is the concentration, ν > 0 is the fluid viscosity and k is the capillaritycoefficient.

Several attempts for the study of regularity have been made. In [1], theauthors study the 2-dimensional case considering an additional diffusion forthe concentration ρ. In [3] the existence (and uniqueness) of regular solutionfor the (KM) model was proved by a fixed point argument of Schauder’stype.

Here, we describe and compare two iterative methods of approximationfor this model. First, we obtain some scheme estimates in regular norms(based on [2]). Secondly, we obtain the convergence from the sequence ofapproximate solutions towards the strong solution of (KM) by means ofCauchy’s argument in weak norms. As a consequence, the existence of reg-ular solutions of (KM) is also proved. By the way, a priori and a posteriorierror estimates are obtained.



References

[1] Kostin, I., Marion, M., Texier-Picard, R. & Volpert, V. A., Modelling of mis-cible liquids with the Korteweg stress, M2AN, 37-5 (2003), 741–753.

[2] Solonnikov, V. A., Estimates of the solutions of the nonstationary Navier-Stokes systems, Tr. Mat. Inst. Akad. Nauk SSSR, 70 (1964) 213–317. Trans-lated in Am. Math. Soc. Transl. Series 2, 75 (1968).

[3] Sy, M., Bresch, D., Guillen-Gonzalez, F., Lemoine, J. & Rodrıguez-Bellido,M. A., Local strong solution for the incompressible Korteweg model.C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 169–174.



Existence and uniqueness results for elliptic problems subject tomixed nonlinear boundary conditions

Jorge Garcıa-Melian1, Julio D. Rossi2, Jose Sabina de Lis*

1Departamento de Analisis Matematico, Universidad de La Laguna, 38271-LaLaguna (Tenerife), Spain; 2 Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, 1428-Buenos Aires, [email protected]


In a nowadays classical paper ([2]) Brezis and Oswald introduced an elegantvariational approach to prove existence and uniqueness of positive solutionsto semilinear Dirichlet problems of the form

∆u = f(x, u) x ∈ Ωu|∂Ω = 0.

We are dealing in this communication with an extension of that problem inwhich the nonlinearity driven by f(x, u) is coupled to boundary conditionsof both Dirichlet and nonlinear Robin type, mixed in a nontrivial way (cf.[1] and the recent work [3] for further results on genuine mixed problems).Specifically, the problem, ∆u = f(x, u) x ∈ Ω

∂u

∂ν= g(x, u) x ∈ Γ, u = 0 x ∈ Γ′,

where Ω ⊂ RN is a bounded and smooth domain, Γ,Γ′ ⊂ Ω are smoothmanifolds with a common smooth boundary, ∂Ω = Γ ∪ Γ′. Assuming thatf(x, u), g(x, u) are Caratheodory functions, we provide necessary and suf-ficient conditions for the existence of a unique solution. More interestingly,such conditions are solely expressed in terms of the main eigenvalues of cer-tain natural associated eigenvalue problems. In this sense, this communica-tion continues previous joint research in nonlinear boundary value problems(cf. [4, 5]).

References

[1] H. Beirao da Veiga, On the W 2,p-regularity for solutions of mixed problems, J.Math. Pures Appl. 53 (1974), 279–290.

[2] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal.10 (1983), 55–64.



[3] E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal. 199 (2003), 468–507.

[4] J. Garcıa-Melian, J. D. Rossi, J. Sabina de Lis, A bifurcation problem governedby the boundary condition I, to appear in NoDEA.

[5] J. Garcıa-Melian, J. D. Rossi, J. Sabina de Lis, A bifurcation problem governedby the boundary condition II, to appear in Proc. London Math. Soc.



Nonlinear double well Schrodinger equations in the semiclassicallimit

Andrea Sacchetti

Dipartimento di Matematica ”G. Vitali”, Universita di Modena e Reggio Emilia,Via Campi 213/B, Modena 41100, [email protected]

2000 Mathematics Subject Classification. 35Q40, 35Bxx, 35K55

We consider a class of time-dependent nonlinear Schrodinger equations

i~∂ψ

∂t=[−~2∆ + V

]ψ + εWψ, ε ∈ R, ψ ∈ L2(Rd),

where V is a symmetric double-well potential and W = W (x, |ψ|) is aperturbation of the type:

– global nonlinear perturbation W (x, |ψ|) = 〈ψ, gψ〉L2g(x), where g(x)is a given function, in any dimension d ≥ 1 [1];

– local nonlinear perturbation W (x, |ψ|) = |ψ|2σ, where σ > 0, in di-mension d = 1, 2 ([2], [3]).

We show that, under certain conditions, the reduction of the time-dependent equation to a two-mode equation gives the dominant term ofthe solution with a precise estimate of the error. More precisely, in thesemiclassical limit we show that the finite dimensional eigenspace associ-ated to the lowest eigenvalues of the linear operator is almost invariantfor times of the order of the beating period and the dominant term of thewavefunction is given by means of the solutions of an exactly solvable finitedimensional dynamical system.

References

[1] V. Grecchi, A. Martinez, A. Sacchetti, Destruction of the beating effect fora non-linear Schrodinger equation. Commun. Math. Phys. 227, pp. 191-209(2002).

[2] A. Sacchetti, Nonlinear time-dependent one-dimensional Schrodinger equationwith double-well potential. SIAM J. Math. Anal. 35, pp. 1160-1176 (2003).

[3] A. Sacchetti, Nonlinear double-well Schrodinger equations in the semiclassicallimit. J. Statist. Phys. 119, pp. 1347-1382 (2005).



Numerical study of stability properties of the Cahn-Hilliardequation

Carola-Bibiane Schonlieb

Faculty of Mathematics, University of Vienna, [email protected],[email protected]

2000 Mathematics Subject Classification. 35K

In [1] we present stability/instability results for the so called kink solutionof the Cahn-Hilliard equation. In particular the behavior of solutions of theCahn-Hilliard equation in a neighborhood of the equilibrium is studied byexploring the Willmore functional.

The Cahn-Hilliard equation is a fourth-order reaction diffusion equationwhich models phase separation and subsequent coarsening of binary alloys,see [2] for a detailed description. The kink solution is a stationary solutionof the equation in one dimension, connecting the two stable equilibria -1 and1 continuously. The analogue in two dimensions are the radially-symmetricbubble solutions, compare [3].

The Willmore functional has its origin in geometry, see [4] for a discus-sion. A generalized version of this functional appears as the first variationof the free energy of the Cahn-Hilliard equation.

Our actual work is motivated by properties of the Willmore functionalasymptotically expanded in the eigenvectors of the linearized Cahn-Hilliardoperator. It can be formally concluded that the Willmore functional de-creases in time for eigenvectors corresponding to a negative eigenvalue andincreases in the case of a positive eigenvalue. Roughly said this means thatlinear instabilities which correspond to positive eigenvalues of the Cahn-Hilliard equation can be detected by considering the evolution of the Will-more functional in time. This behaviour of the Willmore functional seem tobe similar for the nonlinear Cahn-Hilliard operator where of course stabilityneeds further explanation. For nonlinear instabilities this formal conclusionis discussed in various numerical examples.

References

[1] Burger, M., Chu, S.-Y., Markowich, P., Schonlieb, C., Discussion of unstablestates of the Cahn-Hilliard equation. In preparation, 1-19.

[2] Fife, P. C., Models for phase separation and their mathematics, Electron. J.Differ. Equ., Paper No.48, (2000), 26 p.



[3] Alikakos, Nicholas D., Fusco, Giorgio, Slow dynamics for the Cahn-Hilliardequation in higher space dimensions: The motion of bubbles, Arch. Ration.Mech. Anal. 141, No.1, (1998), 1-61.

[4] Willmore, T.J., Riemannian geometry, Oxford Science Publications. Oxford:Clarendon Press. xi, 318 p.(1993).



Decomposition method for fractional partial differentialequations

Nabil T. ShawagfehDepartment of Mathematics, University of Jordan, Amman, [email protected]

2000 Mathematics Subject Classification. 26A33, 35A, 35C, 35G, 35Q

In this paper, Adomian’s decomposition method [1] is effectively imple-mented for solving a class of fractional partial differential equations, wherethe fractional derivative based on Caputo definition. Shawagfeh [2] showedthat the decomposition method is well suited to solve the non linear frac-tional differential equation:

Dαy(x) = f(x, y(x)), x > 0,

where Dα is the Caputo derivative of order α > 0.The aim of the present paper is to implement the Adomain decom-

position method to solve initial value problems for the fractional partialdifferential equation,

Dαt u(x, t) + L[u(x, t)] = F (x, t), t > 0, −∞ < x <∞,

where Dαt is the Caputo fractional derivative, with respect to the variable

t, of order α > 0, L is a partial differential operator on x.The validity of the modified technique is verified through illustrative ex-

amples including Time-fractional diffusion-wave equation, Fractional Fokker-Planck equation and Fractional Black-Scholes equation [3-5]. By this scheme,the solutions are calculated in the from of a convergent power series witheasily computable components.

References

[1] G.Adomian, Solving Frontier Problems of Physics: The Decomposition Method,Kluwer Academic, Dordrecht, 1994.

[2] N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional dif-ferential equations, Appl. Math. & Comp., 131 (2002), 517-529.

[3] A. Hanyga, Multidimensional solutions of time-fractional diffusion-wave equa-tions, Proc. R. Soc. Lond. A, 458 (2002), 933-957.

[4] E. Bakai, Fractional Fokker-Planck equation, solution, and application, Physi-cal Review, 63 (2001), 1-17.

[5] W. Wyss, The fractional Black-Scholes equation, Fract. Calc. Appl. Anal., 3,No 1 (2000), 51-61.



Free Boundary value problems with geometric characteristics onunknown surfaces

Evgeny Shcherbakov

Mathematical Department, Kuban State University, Hidrostroitelei,23,Krasnodar, Russia, [email protected]

2000 Mathematics Subject Classification. 49-XX

The main boundary value problem studied corresponds to the axisymmet-rical cavitational flow and we’ll pose it as as follows:(y−1 Ψx

)x+(y−1 Ψy

)y

= 0, (x , y ) ∈ D, Ψ (x , y ) = 0, (x , y ) ∈ ∂ D,(1)(

2 y 2)−1

|∇Ψ (x , y ) |2 + χH + θK (x , y ) = λ, (x , y ) ∈ Σ, y > h,

(2)

Ψ (x , y ) ∼ y 2

2, (x , y ) →∞ (3)

Here D is the part of meridional section of axisymmetrical domain lyingin the upper half plane which is also symmetrical in order of axis y. Itsboundary consists of two monotone arcs, of unknown curve Σ lying in halfspace y > h connecting two points on the segment (x, y) |y = hwhoseendpoints coincide with those of arcs and the rest part of the segment. Theletters H and K denote mean and Gauss curvature of unknown curve Σrespectively.

We deduce a general representation of functionals defined on Σ whichyield Gauss curvature under its variation and formulate variational principalleading to the existence of generalized solution of the problem (1)–(3). Weprove that this solution is a regular one ([1]).

The method used is further development of the method presented in thepaper [2] and it can be applied for study of generalized minimal surfaces([3]).

References

[1] Chthterbakov, E., Free Boundary Value Problem for Axisymmetric Fluid‘sFlow with Surface Tension and Wedging Forces, Zeitschrift fur Analysis undihre Anwendungen 17 (1998), 937–961.

[2] Garabedian, P.R., Lewy, H., Shiffer, M., Axially Symmetric Navitational Flow,Annals of mathematics 56 (1952), 560–602.



[3] Shcherbakov, E., Variational method of studying generalized Minimal Surfaces,International Conference dedicated to A.N. Kolmogorov (2003), 796.



Elliptic equations with variable anisotropic nonlinearity

S. Antontsev, S. Shmarev*

Departamento de Matematicas Matematica, Universidade da Beira Interior,Covilha, Portugal; Departamento de Matematicas, Univerdidad de Oviedo, [email protected]; [email protected]


We study the Dirichlet problem for the elliptic equations with variableanisotropic nonlinearity

−n∑

i=1

Di

(ai(x, u)|Diu|pi(x)−2Diu

)+ c(x, u)|u|σ(x)−2u = f(x), (1)

−n∑

i=1

Di

(ai(x, u)|u|αi(x)Diu

)+ c(x, u)|u|σ(x)−2u = f(x) in Ω ⊂ Rn. (2)

It is assumed that pi(x), σ(x) ∈ (1,∞), αi(x) ∈ (−1,∞) are given functionssuch that pi(x), σ(x) ∈ C0(Ω) with the logarithmic module of continuity,and that ∂Ω is Lipschitz–continuous. Equations of these types emerge fromthe mathematical modelling of various physical phenomena, e.g., processesof image restoration, flows of electro–rheological fluids, thermistor problem,filtration through inhomogeneous media. We prove that under suitable re-strictions on the coefficients and the nonlinearity exponents the Dirichetproblem for equations (1) and (2) admit a.e. bounded weak solutions whichbelong to the anisotropic analogs of the generalized Sobolev–Orlicz spacesand establish the classes of uniqueness of bounded solutions. The formationof the dead cores is discussed. Using a modification of the method of localenergy estimates [1] we show that unlike nonlinear equations with isotropicdiffusion operator, solutions of equations (1), (2) may identically vanish ona set of nonzero measure either due to a suitable diffusion–absorption bal-ance, or because of strong anisotropy of the diffusion operator. The latteris true even in the absence of the absorption terms. The results extend tocertain equations with the first–order terms and systems of equations of thestructure (2). The presentation follows papers [2, 3].

References

[1] S. N. Antontsev, J. I. Dıaz, S. I. Shmarev Energy Methods for Free BoundaryProblems: Applications to Nonlinear PDEs and Fluid Mechanics. Bikhauser,Boston, 2002. Series: Progress in Nonlinear Differential Equations and TheirApplications, Vol. 48. 329 pp.



[2] S. N. Antontsev, S. I. Shmarev On localization of solutions of elliptic equationswith nonhomogeneous anisotropic degeneracy. Siberian Mathematical Journal,2005. Vol. 46. pp. 765–782.

[3] S. N. Antontsev, S. I. Shmarev Elliptic equations and systems with nonstan-dard growth conditions: existence, uniqueness and localization properties ofsolutions. Nonlinear Analysis Serie A: Theory & Methods (Article in press). 34pp.



Error estimates for a Chernoff scheme for a nonlocal parabolicequation

Moulay Rchid Sidi Ammi

Department of Mathematics, University of Aveiro, 3810-193 Aveiro, [email protected]

2000 Mathematics Subject Classification. 35K15, 35K60, 35J60

The thermistor problem has received a great interest by scientific commu-nity, and has been the subject of a variety of mathematical investigations inthe past decade. We refer in particular to the works of Cimatti, Antontsevand Chipot [1], and the results by various other authors [2] and the refer-ences therein. In this work, we study a general non local parabolic equationwhich, under special simplifications, replaces the classical system of ther-mistor problem [3]. The Chernoff scheme has been studied in particular byMagenes [4], and Verdi and Visitin [5]. This numerical method consist toconstruct a family of time discretization schemes, where at each instant n,is reduced to the resolution of a linear equation with coefficients indepen-dents of n and to a correction which consist to calculate a given function.To addition to stability results, error estimate bounds are established for afamily of time discretization Chernoff scheme.

References

[1] Antontsev. S. N, Chipot. M, The thermistor problem: existence, smoothnessuniqueness, blowup. SIAM J. Math. Anal. 25 no. 4, (1994), 1128–1156.

[2] El Hachimi. A, Sidi Ammi. M. R, Semidiscretization for a nonlocal parabolicproblem, Int. J. Math. Math. Sci. no 10 (2005), 1655–1664.

[3] Lacey. A. A, Thermal runway in a non-local problem modelling ohmic heating,Part I: Model derivation and some special cases. Euro. Jnl of Applied Mathe-matics, vol. 6 (1995), 127-144.

[4] Magenes. E, Remarques sur l’approximation desproblemes paraboliques non lineaires. AnalyseMathematique et Applications, Gautier-Villars, Paris, (1988).

[5] Verdi. C, Visintin. A, Error estimates for a semi-explicit numerical scheme forStefan-type problem. Numer. Math, 52 (1988), 165-195.



Spectral theory of self-adjoint differential vector-operators

Maksim Sokolov

Department of Mechanics and Mathematics, The National University ofUzbekistan, VUZGorodok, Tashkent 700174, Uzbekistan

2000 Mathematics Subject Classification. 34L05, 47B25, 47B37, 47A16

The communication introduces the key notions of the spectral theory ofself-adjoint differential vector-operators (SADVO), which appear to be self-adjoint extensions of minimal operators generated by an Everitt-Markus-Zettl (EMZ) multi-interval differential system (see [1]). The spectral theoryof SADVO was introduced in a series of works, among which we mention [2],[3] and [4]. The main topic of the communication is the method of divisionon subspectra (MDS), which is the principal approach in the constructionof the spectral theory of SADVO. As a demonstration of the MDS, weintroduce the most important constructive theorems of the spectral theoryof SADVO. One of these theorems is the following generalized Mautner-Garding-Brawder theorem (we do not give here the details on the notationsused in the theorem; the theorem is given just to make an idea of the kindof problems discussed):

Theorem [3]. Let T be a SADVO, generated by an EMZ system Ii, τii∈Ω.Let U be an ordered representation of the space L2 = ⊕i∈ΩL

2(Ii) rela-tive to T with the measure θ and the multiplicity sets sk, k = 1,m. Thenthere exist kernels Θk(∨xi, λ), measurable relative to d(∨xi)× θ, such thatΘk(∨xi, λ) = 0 for λ ∈ R \ sk and (⊕i∈Ωτi−λ)Θk(∨xi, λ) = 0 for each fixedλ. Moreover for any bounded Borel set ∆,∫

∆|Θk(∨xi, λ)|2 dθ(λ) ∈ ⊕i∈ΩL

∞(Ini ) ∀n = 1, 2, . . . .

(Uw)k(λ) = limn→∞

∫In

w(∨xi) Θk(∨xi, λ) d(∨xi), w ∈ L2,

where the limit exists in L2(sk, θ). The kernels Θk(∨xi, λ)nk=1, n ≤ m, are

linearly independent as vector-functions of the first variable almost every-where relative to the measure θ on sn.

References

[1] Everitt W.N. and Markus L., Multi-interval linear ordinary boundary valueproblems and complex symplectic algebra, Mem. Am. Math. Soc. 715 (2001).



[2] Sokolov M.S., An abstract approach to some spectral problems of direct sumdifferential operators, Electronic J. Diff. Eq., Vol. 2003, 75, 1-6.

[3] Sokolov M.S., On some spectral properties of operators generated by multi-interval quasi-differential systems, Methods Appl. Anal., 10(4) (2004), 513-532.

[4] Ashurov R.R. and Sokolov M.S., On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space, Appl. Anal. 84(6)(2005), 601–616.



On the dynamics of a Klein-Gordon-Schrodinger type system

Nikolaos M. Stavrakakis

Department of Mathematics, National Technical University, Zografou Campus157 80, Athens, [email protected]

2000 Mathematics Subject Classification. 35B30, 35B40, 35B45, 35B65,35D10, 35L70, 35Q53, 35Q55

We present certain contemporary trends in the theory of Klein-Gordon-Schrodinger type Systems. Then we give some recent results on the exis-tence, uniqueness and asymptotic behaviour of solutions for the followingevolution system of Klein-Gordon-Schrodinger type

iψt + κ∆ψ + iαψ = φψ,

φtt −∆φ+ φ+ λφt = −Reψx,

ψ(v, 0) = ψ0(v), φ(v, 0) = φ0(v), φt(v, 0) = φ1(v),ψ(v, t) = φ(v, t) = 0, v ∈ ∂Ω, t > 0,

where x ∈ Ω, t > 0, κ > 0, α > 0, λ > 0 and Ω is a bounded subset ofRN , with N ≤ 3. This certain system describes the nonlinear interactionbetween high frequency electron waves and low frequency ion plasma wavesin a homogeneous magnetic field. We prove the existence of a global attrac-tor in the strong topology of the space (H1

0 (Ω)∩H2(Ω))2 ×H10 (Ω) which

attracts all bounded sets of (H10 (Ω)∩H2(Ω))2×H1

0 (Ω) in the norm topol-ogy and establish certain energy decay estimates under some parametricrestrictions. Finally, we are going to present some upper estimates for theHausdorff and Fractal dimensions of the global attractor.Acknowledgements. This work was partially supported by a grant from thePythagoras Basic Research Program No. 68/831 of the Ministry of Education ofthe Hellenic Republic.

References

[1] O. Goubet and I. Moise, Attractors for Dissipative Zakharov System, NonlinearAnalysis, TMA, 31, No. 7 (1998), 823-847.

[2] N. Karachalios, N. M. Stavrakakis and P. Xanthopoulos, Parametric EnergyDecay for Dissipative Electron- Ion Plasma Waves, Z. Angew. Math. Phys., 56,(2005), 218-238.

[3] M. N. Poulou and N. M. Stavrakakis, Global Existence and Asymptotic Behav-ior of Solutions for a System of Klein - Gordon - Schrodinger Type, submitted.



[4] M. N. Poulou and N. M. Stavrakakis, Estimates for the Hausdorff and Frac-tal Dimensions of the Strong Global Attractor for a System of Klein-Gordon-Schrodinger Type, in progress.

[5] P. Papadopoulos, M. N. Poulou and N. M. Stavrakakis, Compact Invariant Setsfor a System of Klein-Gordon-Schrodinger Type, in progress.

[6] R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics(2nd Edition), Appl. Math. Sc., 68, Springer-Verlag, New York, 1997.



On a discrete magnetic Laplacian on differential forms

Volodymyr Sushch

Technical University of Koszalin,Sniadeckich 2, 75-453 Koszalin, [email protected]

2000 Mathematics Subject Classification. 35Q60, 39A12, 39A70

Let (M, g) be a Riemannian manifold, dimM = n. Denote Λp(M) the setof all differentiable complex-valued p-forms on M . We define a magneticpotential as a real-valued 1-form A ∈ Λ1(M). We will need a deformeddifferential

dA : L2(M) → L2Λ1(M), ϕ→ dϕ+ iϕA, (1)

where i is the imaginary unit and d is the differential. Let δA be the formaladjoint operator of the operator dA, i.e. δA : L2Λ1(M) → L2(M). Letus identify the magnetic potential A with the multiplication operator A :L2(M) → L2Λ1(M), ϕ→ ϕA. Then the formally adjoint operator δA canbe written as follows

δAω = (δ − iA∗)ω, (2)

where δ, A∗ are the formal adjoint operators of d and A respectively. Using(1),(2), we can write the magnetic Laplacian −∆A ≡ δAdA : L2(M) →L2(M) as follows

−∆Aϕ = −∆ϕ− iA∗dϕ+ iδ(Aϕ) +A∗Aϕ, (3)

where −∆ ≡ δd. Operator (3) is essentially self-adjoint (see [2]).The main purpose of this talk is to construct an intrinsically defined dis-

crete model of the magnetic Laplacian. Speaking about this discrete modelwe do not mean just the corresponding difference operator on a lattice butwe mean a discrete analog of the Riemannian structure on some combina-torial object. We consider discrete forms as certain cochains with complex-valued components. We define a discrete analog of the exterior multiplica-tion operation, a combinatorial Hodge star operator and an inner productof discrete forms. We also construct discrete analogs of the operators (1)-(3)acting on some finite-dimensional Hilbert spaces.

Our approach bases on the formalism proposed by Dezin [1]. In the spiritof [1] we study self-adjointness of the discrete magnetic Laplacian and weproof that the Dirichlet problem for the discrete Poisson type equation hasa unique solution.



One of the formal results is the construction of a nonstandard approx-imation of the generalized solution of the Poisson type equation (with themagnetic Laplacian (3)) under the minimal requirements of smoothness ofthe right hand side, see also [3].

References

[1] Dezin, A. A., Multidimensional Analysis and Discrete Models. CRC Press,Boca Raton, 1995.

[2] Shubin, M., Essential self-adjointness for semi-bounded magnetic Schrodingeroperators on non-compact manifolds. J. Funct. Analysis. 186 (2001), 92–116.

[3] Sushch, V. N., On some discrete model of the magnetic Laplacian. UkranianMath. Bulletin. 2 (2005), 583–599.



Two-scale homogenization for evolutionary variationalinequalities via the energetic formulation

Alexander Mielke, Aida Timofte*

Partial Differential Equations, Weierstraß-Institut fur Angewandte Analysis undStochastik, Mohrenstraße 39, 10117 Berlin, [email protected]

2000 Mathematics Subject Classification. 35B27

This paper is devoted to the homogenization for a class of rate-independentsystems described by the energetic formulation. The associated nonlinearpartial differential system has periodically oscillating coefficients, but hasthe form of a standard evolutionary variational inequality. Thus, the modelapplies to standard linearized elastoplasticity with hardening.Using the recently developed methods of two-scale convergence, periodicunfolding and the new introduced one, periodic folding, we show that thehomogenized problem can be represented as a two-scale limit which is againan energetic formulation, but now involving the macroscopic variable in thephysical domain as well as the microscopic variable in the periodicity cell.

References

[1] Allaire, A., Homogenization and two-scale convergence, SIAM J. Math. Anal.23 (1992), 1482-1518.

[2] Cioranescu, D., Damlamian, A., De Arcangelis, R., Homogenization of nonlin-ear integrals via the periodic unfolding method, C.R. Acad. Sci. Paris Ser. I339 (2004), 77-82.

[3] Mielke, A., Theil, F., Levitas, V., A variational formulation of rate-independentphase transformations using an extremum principle, Arch. Rational Mech.Anal. 162 (2002), 137-177.

[4] Mielke, A., Evolution for rate-independent systems. In Handbook of differentialequations. (ed. by Dafermos, C. and Feireisl, E.). Evolutionary Equations Vol.II, Elsevier B.V. 2005, 461–559.

[5] Nguetseng, G., A general convergence result for a functional related to thetheory of homogenization, SIAM J. Math. Anal. 20 (1989), 608-623.



Distribution of subharmonic and ultraharmonic waves in thepresence of nonlinear mechanism of the energy dissipation

M. Zh. Sergaziyev, A. N. Tyurekhodgaev*

Department of Theoretical and Applied Mechanics, The Kazakh NationalTechnical University named after K.I.Satpaev, 050013 Satpaev str. 22, Almaty,[email protected]

2000 Mathematics Subject Classification. 35L70

The statics and dynamics of mechanical systems with nonlinear mechanismof the energy dissipation which is a contact dry friction is a new scientificdirection in nonlinear mechanics of a deformable firm body.

In case of dynamic deformation in nonlinearity of the dissipation’s mech-anism represents a priori unknown sign-nonlinearity of required function’sspeed. In case of movement an unknown sign function takes plus or a minusunit values and at stops it may take any values between plus and a minusunit and this value should be fixed in task solution process. Such kind oftasks come to consideration of the hyperbolic type equation’s nonlinear sys-tem and are connected with consideration of distribution and attenuationof nonlinear waves.

The first complexity is in the definition of the expression of the fric-tion’s sign-function, which essentially depends on both boundary and ini-tial conditions and its dry friction low. The area of dependence of the deci-sion in such problems is defined by a method of kappa-function of Nikitin-Tyurekhodgaev.

Specificity of these problems is that nonlinearity does not allow to gen-eralize the results of one task to a class of tasks. Analytical results in thedistribution of nonlinear waves in the system with contact dry friction underthe influence of cyclic loads are obtained by the authors.

Based on the results of obtained decisions the following conclusion hasbeen made: there is a class of cyclic loads with the frequency to arbitraryinteger times which differs from the frequency of its own fluctuation ofsystem under the action of which the system will make the establishedsubharmonic or ultraharmonic fluctuations with two frequencies. Besides,one fluctuation coincides with the frequency of its own fluctuation of system,the other - with the frequency of external loading.



References

[1] Tyurekhodgaev, A.N., Nikitin, L.V., Wave propagation and vibration of elasticrods with interfacial frictional slip. In Wave Motion 12 (1990), North-Holland,New York, 513-526

[2] Tyurekhodgaev, A.N., Action of a shock wave in a ground on the pipeline offinal length. In Works of the European conference on geomechanics. Vol. III.Amsterdam 1999, 139-144.

[3] Tyurekhodgaev, A.N., Sergaziyev, M.Zh., Excitation of resonant oscillationsin system with contact dry friction. In Works of the International congress”MechTriboTrans.” Rostov-na-Donu (Russia), 2003, 315-319.



Singularities of Bernoulli free boundaries

Eugen Varvaruca

Department of Mathematical Sciences, University of Bath, Claverton Down, BA27AY Bath, United [email protected]

2000 Mathematics Subject Classification. 35R35, 76B07

A Bernoulli free-boundary problem [4] is one of finding domains in the planeon which a harmonic function simultaneously satisfies linear homogeneousDirichlet and inhomogeneous Neumann boundary conditions. One of theearliest, and still one of the most important, of such problems comes fromBernoulli’s theorem and the constant-pressure condition in the study ofStokes waves in hydrodynamics.

In this talk based on the paper [5] we show that, for a large class ofBernoulli problems, a free boundary which is symmetric with respect toa vertical line through an isolated singular point must necessarily have acorner at that point, and we give a formula for the contained angle. Theassumptions used admit the possibility of other singular points, even un-countably many, on the free boundary. This result is a substantial extensionof that in the theory of hydrodynamic waves [1, 3] on the existence of a cor-ner of 120 at the crest of symmetric Stokes waves of extreme form. We alsoshow that, even in the presence of singularities, any geometrically simpleBernoulli free boundary is necessarily symmetric, extending the result forthe water-wave problem in [2].

References

[1] Amick, C. J., Fraenkel, L. E., Toland, J. F., On the Stokes conjecture for thewave of extreme form, Acta Math., 148 (1982), 193-214.

[2] Okamoto, H., Shoji, M., The Mathematical Theory of Permanent ProgressiveWater Waves, World Scientific, New Jersey-London-Singapore-Hong Kong,2001.

[3] Plotnikov, P. I., A proof of the Stokes conjecture in the theory of surface waves,(In Russian), Dinamika Splosh. Sredy, 57 (1982), 41-76. English translation:Studies in Applied Math. 3 (2002), 217-244.

[4] Shargorodsky, E., Toland, J. F., Bernoulli free-boundary problems, Mem. Amer.Math. Soc., (2006), in press.http://www.maths.bath.ac.uk/∼jft/Papers/memoir.pdf



[5] Varvaruca, E., Singularities of Bernoulli free boundaries, Comm. Partial Dif-ferential Equations, (2006), in press.http://www.maths.bath.ac.uk/∼jft/Papers/ev.pdf



Existence of a Palais-Smale sequence with nonzero limit

Hwai-chiuan Wang

Department of Applied Mathematics, Hsuan Chuang University, Hsinchu, [email protected]


In this talk, We are going to introducing a new method to construct a Palais-Smale Sequence. Moreover, we are going to find a condition for a domainsuch that in which a Palais-Smale Sequence admits a nonzero limit. Thenonzero limit turns out to be a nonzero soluion of the associate semilinearelliptic equation.

References

[1] Hwai-chiuan Wang, Palais-Smale Approaches to Semilinear Elliptic Equationsin Unbounded Domains, Electron. J. Diff. Eqns., Monograph 06, September2004 (142 pages).



Two-fold completeness of root vectors and well-posedness ofhyperbolic problems

Yakov Yakubov

School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, [email protected]

2000 Mathematics Subject Classification. 47A, 34L, 35L

We consider a spectral problem for a system of second order (in the spectralparameter) abstract pencils in a Hilbert space and prove the completenessof a system of eigenvectors and associated vectors; obtain, in some spe-cial cases, the expansion of vectors with respect to eigenvectors; considera relevant application of these abstract results to boundary-value problemsfor second order ordinary differential equations with a quadratic spectralparameter in both the equation and boundary conditions.

We also study an initial boundary-value problem for hyperbolic differential-operator equations and give a relevant application to hyperbolic PDEs withthe same (second) order of differentiation with respect to the time in boththe equation and boundary conditions. For the proof we use some specialtransformations and some results from [1].

For example, the following problem is treated for completeness ques-tions:

L(λ)u := λ2u(x) + λA1u(·)− (b(x)u′(x))′ +B1u(·) = 0, x ∈ [0, 1],

L1(λ)u := αλ2u(0) + u′(0) = 0, L2(λ)u := βλ2u(1) + u′(1) = 0,

where α < 0, β > 0 (note that A1 can be, e.g., a multiplication operatorand B1 can be the first order differential operator with variable continuouscoefficients) and the following problem is treated for hyperbolic PDEs:

L(Dt)u := D2ttu(t, x)+a1(t, x)D2

txu(t, x)+a2(t, x)Dtu(t, x)−Dx(b(x)Dxu(t, x))

+B1(t)u(t, ·) = h(t, x), (t, x) ∈ [0, T ]× [0, 1],

L1(Dt)u := αD2tt[u(t, 0)] +Dxu(t, 0) = h1(t), t ∈ [0, T ],

L2(Dt)u := βD2tt[u(t, 1)] +Dxu(t, 1) = h2(t), t ∈ [0, T ],

u(0, x) = u0(x), Dtu(0, x) = u1(x), x ∈ [0, 1],

where α < 0, β > 0, Dt := ∂∂t , Dx := ∂

∂x . By D2tt[u(t, c)] we mean d2u(t,c)

dt2,

where c = 0, 1.



References

[1] Yakubov, S., Yakubov, Ya., Differential-Operator Equations. Ordinary andPartial Differential Equations. Chapman and Hall/CRC, Boca Raton, 2000.



W 1,p-solvability of partial differential relations

Baisheng Yan

Department of Mathematics, Michigan State University, East Lansing, MI 48824,[email protected]

2000 Mathematics Subject Classification.

We study the solvability of the partial differential relation ∇u(x) ∈ Kfor u ∈ W 1,p(Ω; Rm), where Ω ⊂ Rn is a bounded domain, ∇u(x) is theJacobi matrix of u and K is a given set of m × n matrices. Let U beanother set of m × n matrices. We say that U is W 1,p-reducible to K ifthere exists a constant c(p, U,K) > 0 such that, for some bounded open setΩ ⊂ Rn with |∂Ω| = 0 and for every ξ ∈ U , ε > 0, there exists a functionv ∈ ξx+W 1,p

0 (Ω; Rm) satisfying two conditions: (i) v =∑

i∈N viχΩi , whereΩi is a family of disjoint open subsets of Ω with |Ω \ ∪i∈NΩi| = 0 and vi

satisifes∫Ωi|Dvi(x)|p dx ≤ c(p, U,K) |Ωi| and, either vi|Ωi = ξix + bi with

ξi ∈ U orDvi(x) ∈ K a.e.x ∈ Ωi for each i; (ii)∫Ω dist(Dv(x);K) dx < ε |Ω|.

We say that U is uniformly locally W 1,p-reducible to set K if for each ξ ∈ Uthere exists a bounded set Uξ ⊂ U containing ξ that is W 1,p-reducible toK with constant c(p, Uξ,K) ≤ C(1 + |ξ|p), where C = C(p, U,K) ≥ 1 is auniform constant independent of ξ.

The main result of this paper is the following existence theorem provedin [5].

Main Theorem: Let 1 < p < ∞ and let U be uniformly locally W 1,p-reducible to a closed set K. Then, for any piecewise affine function ϕ ∈W 1,p(Ω; Rm) with Dϕ(x) ∈ U ∪ K a.e. x ∈ Ω and for any ε > 0, thereexists a solution u ∈ W 1,p(Ω; Rm) to the relation ∇u(x) ∈ K a.e. in Ωwhich satisfies

‖u− ϕ‖Lp(Ω) < ε;∫Ω|Du|p ≤ C

(|Ω|+

∫Ω|Dϕ|p

), (1)

where C = C(p, U,K) ≥ 1 is a uniform constant.The method of proof relies on a different Baire’s category argument

from [1] concerning the residual continuity of a Baire-one function used by[2]. Some sufficient conditions for W 1,p-reduction are also given along withcertain generalization of some known results [3] and a specific applicationto the boundary value problem for special weakly quasiregular mappings[4].



References

[1] B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order non-linear systems, J. Funct. Anal., 152 (2) (1998), 404–446.

[2] B. Kirchheim, Deformations with finitely many gradients and stability of qua-siconvex hulls, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 289–294.

[3] S. Muller and V. Sverak, Convex integration with constraints and applicationsto phase transitions and partial differential equations, J. Eur. Math. Soc., 1(4)(1999), 393–422.

[4] B. Yan, A Baire’s category method for Dirichlet problem of quasiregular map-pings, Trans. Amer. Math. Soc., 355(12) (2003), 4755–4765.

[5] B. Yan, On W 1,p-solvability for vectorial Hamilton-Jacobi systems, Bulletin desSciences Mathematiques, 127 (2003), 467–483.



Convergence to equilibrium for nonlinear evolution equations

Songmu Zheng

Institute of Mathematics, Fudan University, Shanghai 200433, [email protected]

2000 Mathematics Subject Classification. 35K40

In a series of papers [1], [2], [3] jointly with H. Wu and M. Grasselli, we haveextended Lojasiewicz-Simon approach to some nonlinear evolution equa-tions with dissipative boundary conditions and obtained the results on con-vergence of the solution to an equilibrium as time goes to infinity. Thesenonlinear evolution equations include:1. The following Cahn-Hilliard equation

∂u

∂t= 4µ,

µ = −∆u− u+ u3, (x, t) ∈ Ω× R+(1)

with dissipative boundary conditionσs∆‖u− ∂νu+ hs − gsu = 1

Γsut, t > 0, x ∈ Γ

∂νµ = 0, t > 0, x ∈ Γ(2)

and the initial condition.2. The following damped semilinear wave equation with critical exponent

utt + ut −∆u+ f(x, u) = 0, (x, t) ∈ Ω× R+ (3)

subject to the dissipative boundary condition

∂νu+ u+ ut = 0, t > 0, x ∈ Γ (4)

and the initial conditions.3. The following hyperbolic=parabolic phase-field system

(θ + χ)t −∆θ = 0, (x, t) ∈ Ω× R+,µχtt + χt −∆χ+ φ(χ)− θ = 0, (x, t) ∈ Ω× R+,

(5)

with dissipative boundary conditions∂νθ = 0, (x, t) ∈ Γ× R+,∂νχ+ χ+ εχt = 0, (x, t) ∈ Γ× R+,

(6)

and the initial conditions.



References

[1] Wu, H., Zheng, S., Convergence to equilibrium for the Cahn-Hilliard equationwith dynamic boundary conditions, J. Diff. Eqs. 204, No. 2(2004), 511–531.

[2] Wu, H., Zheng, S., Convergence to equilibrium for the semilinear wave equa-tion with critical exponent and dissipative boundary condition, to appear inQuarterly of Applied Mathematics.

[3] Wu, H., Grasselli, M., Zheng, S., Convergence to equilibrium for a parabolic-hyperbolic phase-field System with dynamical boundary condition, preprint,2005.



Self-similar solutions for the 1-D Schrodinger map on thehyperbolic plane

Francisco de la Hoz Mendez

Departamento de Matematicas, Universidad del Paıs Vasco, Apdo. 644, 48080Bilbao, [email protected]

2000 Mathematics Subject Classification. 35-99

We study the self-similar solutions developing corners in finite time for thefollowing evolution equation:

Xt = Xs×Xss. (1)

Here, X(s, t) represents a curve parameterized with s and × is the pseudo-cross product according to the Minkowski-metric:

a×b = (a2b3 − a3b2, a3b1 − a1b3,−(a1b2 − a2b1)).

We prove that the solutions corresponding to the initial data

X(s, 0) = A+sχ[0,+∞)(s) + A−sχ(−∞,0](s), (2)

develop a finite-time corner-like singularity. In (2), A+ and A− are vectorsin the unit sphere in R2,1. This problem admits a reformulation in terms ofthe 1-D Schrodinger equation through Hasimoto’s transformation.

An analogous situation has been considered before in the Euclideancase by Gutierrez, Rivas and Vega [3] in connection with the evolutionof vortex filaments. The main difference of our work with respect to theEuclidean case studied by these authors is the proof of the boundedness ofthe generalized trihedron T, e1 and e2 in R2,1.

References

[1] V. Banica, L. Vega, On the Dirac delta as initial conditions for non-linearSchrodinger equations, preprint.

[2] F. de la Hoz, Self-similar solutions for the 1-D Schrodinger map on the hyper-bolic plane, submited to Mathematische Zeitschrift.

[3] S. Gutierrez, J. Rivas, L. Vega, Formation of singularities and self-similarvortex motion under the localized induction approximation, Comm. PDE 28(2003), p. 927-968.



On the local well-posedness for some systems of coupled KdVequations

Borys Alvarez-Samaniego*, Xavier Carvajal

MAB-Universite Bordeaux 1 and CNRS UMR 5466, 351 Cours de la Liberation,33405 Talence Cedex, France; Department of Mathematics, IMECC-UNICAMP,C.P. 6065, CEP 13083-970, Campinas, SP, [email protected]; [email protected]

Using the theory developed by Kenig, Ponce, and Vega, we prove that theHirota-Satsuma system is locally well-posed in Sobolev spaces Hs(R) ×Hs(R) for 3/4 < s ≤ 1. Using the results obtained by Christ, Colliander, andTao, we show ill-posedness for the Hirota-Satsuma system inHs(R)×Hs′(R)for −1 ≤ s < −3/4 and s′ ∈ R. Moreover, we present some commentsto scale changes performed in some previous papers concerning the Gear-Grimshaw system. We introduce some Bourgain-type spaces Xa

s,b for a 6= 0,s, b ∈ R, and we prove some properties for these spaces. Finally, we obtainlocal well-posedness for the Gear-Grimshaw system in Hs(R) ×Hs(R) fors > −3/4, by establishing new mixed-bilinear estimates involving the twoBourgain-type spaces X−α−

s,b and X−α+

s,b adapted to ∂t +α−∂3x and ∂t +α+∂

3x

respectively, where |α+| = |α−| 6= 0.

References

[1] Gear, J. A., Grimshaw, R., Weak and strong interactions between internalsolitary waves, Stud. Appl. Math. 70 (1984), 235–258.

[2] Hirota, R., Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equa-tion, Phys. Lett. A 85 (1981), 407–408.

[3] Kenig, C. E., Ponce, G., Vega, L., A bilinear estimate with applications to theKdV equation, J. Amer.Math. Soc. 9 (1996), 573–603.


on quadratic integral equation of fractional orders - icm 2006

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