Philippe Benilan

SomeL 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions

In this paper we study questions of existence, uniqueness, and continuous dependence for semilinear elliptic equations with nonlinear boundary conditions. In particular, we obtain results concerning the continuous dependence of the solutions on the nonlinearities in the problem, which in turn implies analogous results for a related parabolic problem. Such questions arise naturally in the study of potential theory, flow through porous media, and obstacle problems.

Wiener Regularity and Heat Semigroups on Spaces of Continuous Functions

Let Ω ⊂ ℝ N be open. It is shown that the Dirichlet Laplacian generates a (holomorphic) C o-semigroup on C o(Ω) if and only if Ω is regular in the sense of Wiener. The same result remains true for elliptic operators in divergence form.

Concavity of solutions of the porous medium equation

Abstract : The flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation. It is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables. This bound plays an important role in the mathematical treatment and is related to the entropy of the flow. Since the solutions exhibit interfaces across which v sub x jumps positively, no upper bound is possible globally for v subxx. Nevertheless it is proven that the concavity of v(.,t) in the region where v is positive is preserved in time. This is in itself an interesting geometric property of the solution. It also allows one to obtain precise information about the asymptotic behaviour of the flow.

Sur l’équation générale ut = a (., u , φ (., u ) x ) x + ν dans L 1 : II. Le problème d’évolution

Resume Nous etudions dans cet article l’equation generale ut = a(., u, φ(., u)x)x + v de type parabolique pouvant degenerer en hyperbolique du premier ordre pour certaines valeurs de (x, u). Utilisant la theorie des semi-groupes non lineaires dans L1, nous etablissons des resultats d’existence, d’unicite et de dependance continue par rapport aux donnees, d’une « bonne solution » du probleme de Cauchy ou de problemes aux limites associes a cette equation sous des hypotheses tres generales sur les donnees. Avec des hypotheses complementaires, nous montrons que cette « bonne solution » est « solution entropique », nous etudions l’unicite des solutions faibles et l’existence de solution forte.

Existence of attractors in L∞ (Ω) for a class of reaction-diffusion systems

Let us consider as an example, the reaction-diffusion system named “Brusselator”:

L^1

{u_t} - {d_1}\Delta u = {u^2}v - \left( {B + 1} \right)u + A in \left( {0,T} \right) \times \Omega

Solutions of the Porous Medium Equation in R(N) under Optimal Conditions on Initial Values.

(0.1)