The Analysis Of The Anisotropic Parabolic Equations

This paper studies the anisotropic parabolic equations,the existence of local solu-tion is obtained under the framework of anisotropic variable exponent Sobolev space.Then the main contents of this thesis is to prove the existence of weak solutions of anisotropic nonlinear parabolic equations with convection terms by the re-normalized solution method.Finally,the extinction,positivity and blow-up of the solutions of the anisotropic variable exponent equations are discussed.In chapter 1,we mainly introduce the background and the current states and progress of anisotropic partial differential equations,give the basic definition and prop-erties of the related spaces,and list the main contents of the structure of this thesis.In chapter 2,we deal with the following anisotropic variable parabolic equationby the assumption of 2<Pi(x)<N,u0?C(?),f(s)?C(R),and there is nonnegative functio h(s)?C1(R),such that |f(s)?h(s),the local existence of solutions of problem with the homogeneous Dirichlet boundary conditions is obtained.Moreover,the blow-up principles of weak solutions with positive initial energy is proved by the method of energy functional.In chapter 3,we prove the extinction and positivity for the solution of the anisotrop-ic evolution equationAt the same time,we consider the blow-up of the solution of the following equationwhere 2<pi<N,m?1.Besides,we assume f(s)?C(R),and there is a non-negative function h(s)?C1(R)makes |mu m-1 f(u)|?h(um).In chapter 4,we concern with the initial-boundary value problem of the following nonlinear parabolic equationUnder suitable conditions on Bi(x,t),Pi(x)for i=1,2,...,N,F and ?,employing the De Giorgi iteration technique,we establish a priori L?-estimate and thus prove the existence of solutions for the above problem in the anisotropic variable exponent space.