Study On The Blow-up Problems For Two Classes Of Nonlinear Reaction Diffusion Equations

The purpose of this paper is to discuss two problems,one of which is the blow-up solutions and global solutions for a class of nonlinear reaction diffusion equationsunder Dirichlet boundary conditions,and the other is the blow-up solutions for a classof nonlinear reaction diffusion equations with Neumann boundary conditions.Themain methods we employed in our discussions are auxiliary function method and thefirst-order differential inequality technique.This paper includes three chapters.In chapter 1,firstly,we briefly summarize thee research background,significanceand the research progress of the blow-up problems for nonlinear reaction diffusion equa-tions.We then present maximum principles for nonlinear reaction diffusion equationsand some fundamental inequalities,which are applied in this paper.In chapter 2,we consider the following problem:where ?(?)RN(N?2)is a bounded domain with smooth boundary(?)?.By con-structing auxiliary functions and using the first-order differential inequality technique,we give the sufficient conditions for the existence of the blow-up solution,the sufficientconditions for the global existence of the solution,an upper bound for the "blow-uptime",and an lower bound for the "blow-up time".In chapter 3,we study the following problem:where ?(?)RN(N?2)is a bounded convex domain with smooth boundary(?)?.Under appropriate assumptions on the functions f,?,k,g and u0,a lower bound on blow-up time was showed by applying a differential technique when blow-up occurs.Moreover,the conditions which imply that,blow-up occurs are obtained.