The Research About The Solutions Of A Class Of P(x)-Laplacian Evolution Equations

This thesis mainly studies the existence and uniqueness of weak solutions to a class of p?x?-Laplacian evolutionary equations.The full text consists of three parts.The first chapter reviews and summarizes the main progress and the background of some nonlinear parabolic equations.Then,we give the definition of space L^p?x????,W^1,p?x????,and some important lemmas.Finally,we introduce the main structure of this article.In second chapter,we consider the existence and uniqueness of weak solutions about the initial boundary value problem of the nonlinear parabolic of p?x?-Laplacian evolutionary equation.This chapter uses the difference method to transform the parabolic problem to a sequence of elliptic problems and proves the existence of so-lutions with p?x?satisfying 2N/?N+2??p?x?<2 but without the logarithmiac Holder continuity condition.The main conclusions of the chapter are as following:Since u₀?L^?????W₀^1,p?x????,2>p⁺ = ess supp?x??ess infp?x?>2N/?N+2? and f?x,t,z??C¹????x[0,T]×R?,|f?x,t,z?|?C₀???x,t?+|z|^??.When?<p^-?1 or ?=p^-?1 and |?| is sufficiently small,where |?| denotes the Lebesgue measure of?,then there exists a unique weak solution of the above question such thatu?L^??Q_T??L^?(0,T;W₀^1,p?x????),u_t?L²??×?0,T??.The third chapter discusses the existence and uniqueness of the solutions of the boundary degeneracy p?x?-Laplace equation with absorption term.The main con-clusions of the chapter are as following:If p^->1,u₀?L^????,?^?|?u₀|^p+?L¹???,?1?While ?<p^-?1,then the equation admits a unique solution with the fixed initial boundary value conditions.?2?While ?>p⁺?1,then the solution of equation is completely controlled by the given initial value.