Rothe time-discretization method for a nonlinear parabolic problem in Orlicz-Sobolev spaces

Abstract

In this paper, we prove the existence and uniqueness of entropy solutions for the following equations in Orlicz spaces: ∂u∂t-div(ax,∇u(x,t))+β(u)=finQT=]0.T[×Ωu=0onΣT=]0.T[×∂Ωu(0,.)=0inΩ,(1)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{c} \\frac{\\partial u}{\\partial t}-div\\Big (a \\left( x,\\nabla u(x,t)\\right) \\Big )+ \\beta (u)=f\\text { in }Q_{T}= ] 0.T[ \\times \\Omega \\\\ u=0\\text { on }\\Sigma _{T}=] 0.T[ \\times \\partial \\Omega \\\\ u(0,.)=0 \\text { in }\\Omega , \\end{array} \\right. \\qquad (1) \\end{aligned}$$\\end{document}where f is an element of L^{1}( Q_{T} ), the term -text{ div }Big (a(x,nabla u(x,t))Big ) is a Leray-Lions operator on W_0^{1,x}L_M(Omega ), with M(.) does not satisfy the Delta _2 condition and beta is a continuous non decreasing real function defined on mathbb {R} with beta (0)=0. The investigation is made by approximation of the Rothe method which is based on a semi-discretization of the given problem with respect to the time variable.