Analysis Of A Free Boundary Problem Modeling The Granuloma Formation

In this paper we study a free boundary problem based on the mathematical model of the granuloma formation.This model is used to describe the growth of granuloma and is of great significance for the treatment of visceral leishmaniasis.The mathematical model consists of coupled parabolic equations and hyperbolic e-quations with moving boundary,and it describes the interaction between dendritic cells,macrophages,T cells,the leishmania and the cytokineses involved in the granuloma for-mation.We study the existence and uniqueness of the global solution of this problem.This paper is divided into five chapters:In Chapter 1,we introduce the background of the mathematical model and sketch the main result and method of this paper.In Chapter 2,we introduce the relevant notations and basic lemmas needed in the research process.In Chapter 3,we use a proper transformation to convert the free boundary problem into an initial boundary problem on a fixed domain.In Chapter 4,we study the existence and uniqueness of the local solution of this prob-lem.Firstly,we introduce a reasonable metric space and an appropriate mapping.Then we use thetheory of parabolic equations,characteristic theory of hyperbolic equations and Banach fixed point theorem to prove the local existence and uniqueness.In Chapter 5,we derive the a priori estimate and then obtain the existence and uniqueness of the global solution by extension method.