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

摘要

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} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} left{ begin{array}{c} frac{partial u}{partial t}-divBig (a left( x,nabla u(x,t)right) Big )+ beta (u)=ftext { in }Q_{T}= ] 0.T[ times Omega u=0text { on }Sigma _{T}=] 0.T[ times partial Omega u(0,.)=0 text { in }Omega , end{array} right. qquad (1) end{aligned}$$end{document}where fdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ f$$end{document} is an element of L1(QT)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^{1}( Q_{T} )$$end{document}, the term -div(a(x,?u(x,t)))documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$-text{ div }Big (a(x,nabla u(x,t))Big )$$end{document} is a Leray-Lions operator on W01,xLM(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W_0^{1,x}L_M(Omega )$$end{document}, with M(.) does not satisfy the Δ2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Delta _2$$end{document} condition and βdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$beta$$end{document} is a continuous non decreasing real function defined on Rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}$$end{document} with β(0)=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$beta (0)=0$$end{document}. 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.