Weak Solutions and Numerical Blow-up for the p(x)-Laplacian Equation

Hamid El Ouardi

Hamid El Ouardi Research laboratory in Engineering (LRI), University Hassan II of Casablanca, National Higher School of Electricity and Mechanics, Equipe S�curit�, Systemes & Modelisations, ENSEM, BP. 8118, Km7, Morocco.

Keywords: p(x)-laplacian equation, existence, uniqueness, numerical blow-up, variable exponents, difference method

Abstract

This paper aims to investigate in the evolution equation p(x)- Laplacian with the initial boundary value question for the : a(x) ?u) = f(x, u). The paper translates the parabolic equation into the elliptic equation by using a ?nite difference method, and then the existence and uniqueness solution are obtained. The blow-up property is shown, by using the energy method. Matlab (Ode45 subroutine) was used for some numerical experiments just to illustrate the general results. The generalization of this study to nonlinear parabolic systems of (p ?u ?t ? div(|?u| p(x)?2 (x))-Laplacian type : a i (x)u it ? div(|?u i | p i (x)?2 ?u i ) = f i (x, u 1 , u 2 1 (x), p ), (i = 1, 2) has been established in another work. The results recover and extend some known results on the existence and long-time behavior of solutions to the semilinear heat equation and the p- Laplacian equations.