Some Studies On Free Boundary Problems For Chemotaxis Models And Fractional Chemotaxis Models

In this thesis, we consider the existence theory of the chemotaxis system which consists of a class of partial differential equations arising in mathematical biology.This thesis is divided into four parts.In Part 1, we first recall the biological background of the system, the history of Keller-Segel model and free boundary problem. Then we summarize some known results. Based on this, we describe our main results.In Part 2, we consider free boundary problems for parabolic-elliptic, fully parabol-ic and parabolic-hyperbolic type chemotaxis models. In Chapter 2 and Chapter 3, we introduce an extra equation with variable coefficients describing the free boundary and use the contraction mapping principle and some estimates on elliptic and parabol-ic equations to establish the existence of the solution for parabolic-elliptic and fully parabolic type chemotaxis models in high dimensional symmetric domain respectively. Then the existence of the solution for parabolic-hyperbolic type chemotaxis model with fixed boundary and variable coefficients free boundary is discussed by the contraction mapping principle and some estimates on parabolic and hyperbolic equations in the following Chapter 4.In Part 3, the existence of the solution for the fractional chemotaxis model in Sobolev spaces and Besov spaces is considered. Based on the difference of the initial data, we use the contraction mapping principle and approximate methods to establish the existence of the solution in Sobolev spaces for such kind of model in Chapter 5. In Chapter 6, the existence of the solution in Besov spaces is discussed by approximate methods.In Part 4, we summarize the main results of the thesis and propose some interesting problems to be done.