Research On One Class Of Singularly Perturbed Reaction Diffusion Equation With Discontinuity Term

In the present dissertation,a kind of singularly perturbed reaction-diffusion prob-lem with discontinuous reaction and diffusion terms is studied.According to the sin-gular perturbation theory and contrast structure theory,a uniformly valid asymptotic solution for such problem is constructed via the standard Vasil'eva's boundary layer function method.In the meantime,the estimation of remainder and the analysis of sta-bility are presented.This dissertation is divided into three chapters and organized as follows:In the first chapter,the research background and progress of singular perturbation are introduced,and the fundamental method as well as some useful theorems which plays important roles in the future research are reviewed.The main work of this thesis will be presented in the end of this chapter.Thereafter in chapter 2,the problem for a singularly perturbed reaction-diffusion systems subject to Dirichlet boundary value conditions with discontinuous reaction and diffusion terms is considered.The formal asymptotic solution with internal and bound-ary layers will be constructed by using Vasil'eva's method.It can be observed that the internal layer is located in a small neighborhood of the curve for the discontinuous re-active term.The smoothness of the asymptotic solution is guaranteed by contrast struc-ture theory and sewing method.The existence of a periodic solution which is locally asymptotically stable with stability region O(?)in the above small neighborhood,and the accuracy order of the asymptotic expansion are obtained by differential inequality method.The summary and the significance as well as prospect of this paper will be provided in the last chapter of this paper.