Upper Bound Of Blow Up Time For The Solutions Of Reaction Diffusion Equations With Special Diffusion Processes

In this paper,we consider the following initial boundary value problem of a reaction diffusion equation with special diffusion processes where Ω is a bounded domain of RN(N ≥ 3)with smooth boundary(?)Ω,0∈Ω，1＜p＜(?)and uo(x)∈H01(Ω).This thesis is organized as follows.In Chapter 1,we briefly recall some backgrounds of the problem under consideration and overview some related works that are concerned with blow-up phenomena of evolution equations.In Chapter 2,as preliminaries,we present the definitions of weak solutions and functionals I(u),J(u),investigate their basic properties and introduce two necessary lemmas.The main results of this paper are also stated in this chapter.In Chapter 3,by using Hardy inequality,we obtain a new criterion for the solution of Problem(0.1)to blow up in finite time.Moreover,the upper bound of the blow-up time T*is estimated.The main result of this thesis is summarized as follows.Theorem Assume that u is a weak solution of Problem(0.1)with initial data u0 H01(Ω).Suppose that one of the following statements holds:(i)J(u0)<0;(ii)0≤(u0)＜(?),where C*=(?)(CN is a constant of the Hardy inequality),then T*<+∞,which means that u blows up in finite time.Moreover,the upper bound of T*is estimated as follows:In case(i),(?)In case(ii),(?).