Techniques de construction des fonctionnelles de Lyapunov en gradient pour une classe des systèmes de réaction-diffusion.

Abstract

The aim of this work is to present techniques of construction of Lyapunov functional in gradient allowing to prove the global existence in time for solutions of systems which are formed by partial differential equations of parabolic type calls Systems of reaction-diffusion equations. It is well known that to show the global existence, it suffices to show, by using the regularizing effect (see D. Henry [14]) that the term of reaction is in L(0,Taxi L) for a certain pon 2. In the case where the reaction is of exponential growth with an exhibitor that is enough sufficient, the elementary Lyapunov functional (polynomials, ...)do not give the global existence (see S. Kouachi [24] and S. Kouachi & A. Youkana [25]). The functional which used here can give priori estimations of the solution in the spaces L2(0,TMx: I'), then by applications of Sobolev injections we can easily deduce the solution is in the space L" (0,TMax; L') for a certain p>n/2 that gives the global existence of the solution by the regularizing effect's principle.