Research On The Initial-boundary Value Problems For A Class Of Nonlinear Degenerate Evolution Equations

In this thesis,we study the initial-boundary value problems for a class of nonlin-ear degenerate evolution equations,including the global existence,decay estimate and blow-up of solutions for finitely degenerate semilinear parabolic equations and pseudo-parabolic equations,and the global existence,decay estimate and blow-up of solutions for infinitely degenerate semilinear parabolic equations and pseudo-parabolic equations with logarithmic nonlinearity.This article is divided into five chapters as followsIn Chapter 1,we first introduce the sources and research status of the problems for both parabolic equations and pseudo-parabolic equations,and then review the history and development of degenerate elliptic operator.Finally,the main results of this article and preliminaries are givenIn Chapter 2 and 3,we study the global existence,decay estimate and blow-up of solutions to the initial-boundary value problem for the following finitely degenerate semilinear parabolic equations ut-△Xu=|u|p-1u,x∈Ω,t>0.and pseudo-parabolic equations ut-△Xut-△Xu=|u|p-1u,x∈Ω,t>0.respectively.Here,X=(X1,X2,…,Xm)is a finitely degenerate system of vector fields,Xj*=-Xj,△X=∑j=1 m Xj2 in a finitely degenerate elliptic operator,1<p<v+1/v-2,and v is the generalized Metivier index of X on bounded open domain C Rn.Firstly,we define a family of potential wells,discuss their properties,and show the invariant set of solutions of corresponding problems.Then,combining Galerkin method,we obtain the existence and asymptotic behavior of the global weak solutions for both problems above,respectively.Finally,using concavity method and properties of a family of potential wells,we obtain blow-up in finite time of solutions for corresponding problems when the initial value in unstable set.In Chapter 4 and 5,we study the global existence,decay estimate and blow-up of solutions to the initial-boundary value problem for the following infinitely degenerate semilinear parabolic equations ut-△Yu=ulog |u|,x∈Ω,t>0,and pseudo-parabolic equation ut-△Yut-△Yu=ulog |u|,x∈Ω,t>0 with logarithmic nonlinearity,respectively.Here,Y=(Y1,Y2,…,Ym)is an infinitely degenerate system of vector fields,Yj*=-Yj,and △Y=∑j=1m Yj2 is an infinitely degenerate elliptic operator.Firstly,we redefine a family of potential wells,discuss their properties,and obtain the invariant set of solutions of corresponding problems.Moreover,by Galerkin method and the logarithmic Sobolev inequality,we obtain the existence and asymptotic behavior of the global weak solutions for both problems above,respectively.Finally,using the properties of a family of potential wells,we prove blow-up at +∞ of solutions for corresponding problems when the initial value in unstable set.