Evolutionary problems with quasilinear elliptic operators TITLE AND ABSTRACTS. A parabolic antimaximum principle

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1 Evolutionary problems with quasilinear elliptic operators Special Session, WCNA-2004 organized by G. Hetzer and P. Takáč TITLE AND ABSTRACTS A parabolic antimaximum principle J. Fleckinger-Pellé - joint work with J.I.Diaz - Jacqueline Fleckinger-Pellé CEREMATH UMR MIP, Université Toulouse 1, Place A.France, F Toulouse Cedex, France, jfleck@univ-tlse1.fr Abstract. We adapt the antimaximum principle valid for elliptic problems to the parabolic ones. We study the positivity, for large time, of the solutions to the heat equation Q a (f, u 0 ): t u u = au + f(t, x), in Q =]0, [ Ω, Q a (f, u 0 ) u(t, x) = 0 (t, x) ]0, [ Ω, u(0, x) = u 0 (x), x Ω, where Ω is a smooth bounded domain in R N and a R. Here the data u 0 and f are not necessarily of the same sign.

2 On the stability of low-energy equilibria for a class of nonlinear reaction-diffusion equations M. Lazzo, Department of Mathematics, University of Bari, via Orabona 4, I Bari, Italy (joint work with P.G. Schmidt, Auburn University, USA) Abstract We consider a class of nonlinear reaction-diffusion equations in R n, whose low- -energy equilibria exhibit a concentration behavior. For example, the prototype equation u t div ( a(x) u ) + λ u = u p 2 u, with subcritical exponent p, has stationary solutions that concentrate around global minima of the diffusion coefficient, provided that lim inf x a(x) > inf x Rn a(x) > 0. We study the stability properties of such equilibria.

3 Large-time Behavior of Degenerate Nonlinear Diffusion-reaction Problems Monique Madaune-Tort Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l Adour, France monique.madaune-tort@univ-pau.fr We consider an initial-boundary value problem for the degenerate nonlinear diffusion equation (1) u t ϕ(u) + div( P f(u)) + g(x, u) = 0, t ]0, + [, x Ω where Ω is a bounded regular domain in R N. The datum P belongs to W 1, (Ω), ϕ is an increasing C 1 -function on [0, + [ such that ϕ (0) = 0, the functions f and g are locally Lipschitz continuous on [0, + [. Equation (1) is allowed to degenerate, that is to say there exist values of v for which ϕ (v) = 0. Therefore, the solution u may lose regularity in some regions of the domain Ω. Our goal is to study the large-time behavior of global in time weak solutions of (1). So, we suppose that the growth rate of g is less than the one of ϕ, that is to say (2) α [0, 1[, (A, K) (R +) 2 ; (x, v) Ω ]A, + [ K 0, g(x, v) Kϕ(v) α. It is already known that without Hypothesis (2) blow-up may occur in finite time even in the borderline case α = 1 [4]. Equation (1) arises in several areas of science, for instance in models for gas flow in a porous medium or for population dynamics. In these both situations a classical choice of the function ϕ is given by ϕ(u) = u m, m 1. Then, we may have g (u) = cu p when u is a gas density or g (u) = c (1 u) u when u is the density of a biological population. There is an extensive literature about the large-time behavior of solutions to strictly parabolic problems. But for degenerate equations most of the papers giving a convergence result are about the one spatial dimensional case. So the problem of convergence for N 2 remains open except in some particular cases [2], [3]. In this talk, we first justify under Hypothesis (2) that for any nonnegative bounded initial state Equation (1) has a nonnegative bounded global solution u. Then, the asymptotic behavior of u depends on the properties of the function v g (x, ϕ 1 (v)). When for a.e. x Ω v g (x, ϕ 1 (v)) /ϕ 1 (v) is increasing on ]0, + [, we can deduce from a result of [1] about elliptic equations that the ω-limit set is a singleton. Moreover we prove that when Ω is a ball the limit state for the equation u t ϕ(u) + g(u) = 0

4 is a radially symmetrical function. The special case when v g (x, ϕ 1 (v)) is locally Lipschitz with respect to v is also studied. References [1] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis, Theory, Methods & Applications, 10, n o 1, 55-64, [2] M. Escobedo, E. Feireisl, P. Laurençot, Large time behaviour for degenerate parabolic equations with dominating convective term, Commun. Partial Differ. Equations 25, n o 1-2, 73-99, [3] E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Differential Equations, J. Dyn. Differ. Equations 12, n o 3, , [4] H.A. Levine and P.E. Sacks, Some Existence and Nonexistence Theorems for Solutions of Degenerate Parabolic Equations, J. Differential Equations, 52, , 1984.

5 On a singular semilinear equation arising in the sharp problem of the ideal MHD J.F. Padial Univ. Politécnica de Madrid (Spain) Dpto. de Matemática Aplicada E.T.S. de Arquitectura jfpadial@aq.upm.es Joint work with J.I. Díaz and J.M. Rakotoson Universidad Complutense de Madrid (Spain) and Université de Poitiers (France) The sharp problem of the ideal MHD is characterized by a piecewise constant pressure. By reducing the problem to the two-dimensional case, this type of problems can be formulated in terms of a singular semilinear equation for the current function. More in general, the formulation arises also as a particular case of the Bernoulli problem and can be stated as follows: to find a function u : Ω R N R and a subset A Ω such that (B) u = 0 in Ω \ A, u = 0 on Ω, u = 1 on A, u n = q on A, with q a positive given number. Here Ω denotes a bounded open subset of R N, N 2. We show that the problem can be reformulated, under suitable assumptions, in terms of some nonlocal problems (involving the nondecreassing rearrangement) of the following type: to find u C ( Ω) H 1 0 (Ω) such that (B 1 ) Ω u (x) ϕ (x) dx = qu (0) (u 1 (1)) ϕdh N 1, for all ϕ C ( Ω) H 1 0 (Ω), u (0) = 1. By (u 1 ) we denote the boundary of the set {x Ω : u (x) = 1} and u (0) = max Ω (u). We study the existence and non existence and the uniqueness of solutions for the one dimensional case, the N dimensional radial case and the case of a non-radially symmetric domain Ω R N but satisfying some special topology properties.

6 On Nonlinear Biot s Consolidation Models Patrick Saint-Macary Université de Pau et des Pays de l Adour Laboratoire de Mathématiques Appliquées, IPRA Avenue de l Université, BP Pau Cedex, FRANCE patrick.saint-macary@univ-pau.fr Wave propagation in fluid saturated porous media can be used in various domains like petroleum geophysics or medicine as a non-invasive tool for imaging. Classically, this phenomenon is described by a model due to M.A. Biot [1] which consists of a coupled system of mixed hyperbolic-parabolic equations where the unknowns are the structure displacement vector field u and the fluid pressure p: ρ(x) 2 u t 2 (λ (x) div u) ((λ(x) + µ(x))div u) t (3) div (µ(x) u q 2 u) + α p = f c 0 (x) p t + α div u t div (k(x) p) = h. First equation combines Hooke law for elastic deformations with the balance momentum equations to describe the time evolution of u. As far as the fluid pressure is concerned, p satisfies a diffusion equation taking the Darcy law for laminar flows into account. The coupling terms with α traduce the pressure-deformation effects due to the fluid-structure interactions, the so-called consolidation effects. When λ > 0, secondary consolidation phenomenon is also considered while, when λ = 0, System (3) describes thermoelastic phenomena where p is the temperature. Another interesting limit case to the Biot model is the quasi-static system arising when the density of the structure ρ is negligible. The physical leading works of M.A. Biot [2] and K. Terzaghi [5] have given rise to mathematical extensions. Among them, C.M. Dafermos developed a rigorous theoretical study for the linear thermoelastic problem (λ = 0, q = 2) constructing strong solutions [3]. Next, the linear quasi-static system (ρ = λ = 0, q = 2) was studied by R.E. Showalter in [4] using semi-group theory. Herein, we are interested in the Biot model and in the quasi-static case when the fluid within the structure can be non-newtonian (q 2). We focus on these systems in the one-dimensional case and we prove existence-uniqueness results before clarifying how the general model provides an approximation of the quasi-static problem by estimating the approximation error as a function of ρ. References [1] M.A. Biot: General theory of three-dimensional consolidation, J. Appl. Phys. 12, , (1941). [2] M.A. Biot: Acoustics, elasticity and thermodynamics of porous media: twenty-one papers by M. A. Biot, Ivan Tolstoy Ed., (Springer-Verlag, New-York, 1992).

7 [3] C.M. Dafermos: On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29, , (1968). [4] R.E. Showalter: Diffusion in poro-elastic media, Jour. Math. Anal. Appl. 251, , (2000). [5] K. Terzaghi: Erdbaumechanik auf bodenphysikalisher grundlage, Leipzig F. Deuticke, (1925).

8 Explosive Behavior in a Class of Reaction-Diffusion Systems Arising from Fluid Dynamics P.G. Schmidt, Department of Mathematics, Auburn University, AL , USA (joint work with J.I. Diaz, Madrid, and M. Lazzo, Bari) Abstract Thermally driven flows of viscous fluids are governed by balance equations for momentum, mass, and energy. Employing the so-called Boussinesq approximation, one is led to the Navier-Stokes equations for a viscous incompressible fluid, coupled to a heat equation. If viscous heating (that is, heat production due to internal friction) is neglected, the associated initial-value and boundary-value problems are well posed in the same sense as for the classical Navier-Stokes system. This may not be the case if viscous heating is taken into account. In the present paper, we show that a class of simpler, yet closely related reaction-diffusion systems exhibits explosive behavior: finite-time blow-up in the time-dependent case, boundary blow-up in the stationary case.

9 Stationary profiles of degenerate problems with inhomogeneous saturation values Shingo Takeuchi (Kogakuin University, Japan) This work is concerned with the profiles of non-negative solutions for stationary problems between the degenerate diffusion by p-laplacian and a reaction with inhomogeneous saturation value. It is shown that, if a parameter for diffusion is sufficiently small, then the solution attains the saturation value of reaction in each region where the value is constant. We also make mention of the inverse problem for this and the associated non-stationary problems. This work was supported by MEXT, Grant-in-Aid for Young Scientists (B), No

10 Stabilization of solutions in a climate model involving the p-laplacian L. TELLO Universidad Politécnica de Madrid, E.T.S. Arquitectura, Dept. Matemática Aplicada, Av. Juan de Herrera, Madrid, Spain, ltello@aq.upm.es We are concerned with the mathematical treatment of a nonlinear model for the coupling of the mean surface temperature of the Earth with the ocean temperature. The model incorporates a dynamic and diffusive boundary condition. The diffusion at the boundary is given by the p-laplacian operator. Moreover the boundary condition includes the Coalbedo function (a bounded maximal monotone graph). Our purpose is to study the stabilization of solutions of the evolution model as time tends to infinity.

11 Existence and Uniqueness of Solutions for Complex Ginzburg-Landau Equations Noboru Okazawa and Tomomi Yokota Department of Mathematics, Science University of Tokyo 26 Wakamiya-cho, Shinjuku-ku, Tokyo , Japan Let Ω be a bounded domain in R N (N N) with C 2 -boundary Ω. We consider the following initial-boundary value problem for the complex Ginzburg-Landau equation: (CGL) u t (λ + iα) u + (κ + iβ) u q 2 u γu = 0 in Ω R +, u = 0 on Ω R +, u(x, 0) = u 0 (x), x Ω, where λ, κ R + := (0, ), α, β, γ R and q 2 are constants, and u is a complexvalued unknown function. The existence and uniqueness of global strong solutions to (CGL) with u 0 L 2 (Ω) (smoothing effect on the initial data) have already been proved by ourselves (2002) under the condition (4) κ 1 β [0, c q ], c q := 2 q 1/(q 2) without any restriction on q 2 (monotonicity methods). In this talk we shall show that if we impose an additional condition on q, then we can establish the smoothing effect of (CGL) on the initial data even if condition (4) breaks down. Main Theorem. Let N N, λ, κ R +, α, β, γ R and 2 q 2 + 4/N. Then for any u 0 L 2 (Ω) there exists a unique global strong solution u( ) C([0, ); L 2 (Ω)) to (CGL) such that u( ) C 0,1/2 loc (R + ; L 2 (Ω)) C(R + ; H0(Ω)), 1 du dt ( ), u( ), u q 2 u L 2 loc(r + ; L 2 (Ω)), u(t) L 2 e γt u 0 L 2 t 0, u(t) v(t) L 2 e K 1t+K 2 e 2γ + t ( u 0 2 L 2 v 0 2 L 2) u 0 v 0 L 2 t 0, where v( ) is a unique strong solution to (CGL) with v(0) = v 0 L 2 (Ω), γ + := max{γ, 0}, and K 1 and K 2 are positive constants depending only on λ, κ, β, γ, q, N. Moreover, we would like to discuss the quasilinear problem (CGL) with replaced with the p-laplacian p defined as p u := div( u p 2 u).