On The Qualitative Behavior Of Solutions For A Class Of Semilinear Parabolic Partial Differential Equations

This dissertation deal with various properties of the solutions of a class of initial-boundary value problems for a special semilinear parabolic equation under a kind of Dirichlet boundary condition.Semilinear parabolic equation is the simplest generalization of classical heat equation,now there is a large number of research literatures concerning about the properties of solutions of semilinear parabolic equations.This paper focus on the local time existence and asymptotic behavior of the solution under such conditions according to the different exponents,several methonds are put foward and we give priori estimates and qualitative analysis for more general situation.The thesis is composed of six chapters.In Chapter 1,we summarize the background of the related issues and state the outline of the present thesis.In Chapter 2,we state the background knowledge used in this thesis,mainly the basic concepts and theorems in functional analysis and Sobolev space.In Chapter 3,we mainly study the local-time existence and uniqueness of classical solution.In Chapter 4,we investgate the long-time extenability of maximal solution,under the condition 0<p≤1,we discuss the long-time existence of the positive classical solutions corresponding to Cauchy problem and Dirichlet’s boundary value problem,obtain the sufficient condition for the solution to blow up in finite time when p>1.In Chapter 5,we study the long-time existence and asymptotic behavior of solution under small initial values.In Chapter 6,we put forward the boderline weak solution theory in an organized way,including priori estimates of long-time classical solutions and the construction of boderline weak solutions.