The Mathematical Analysis Of Biological Chemotaxis Models

In this thesis, we mainly discusses a class of bio-mathematical model for chemotaxis, by using partial differential equation theory we give some qualitative mathematic analysis of some model and the analysis results have a certain amount of guidance form biological view.In recent years, the mathematical models for biological research interest in increasing year by year. Since 1970 Keller and Segel proposed the classic model of chemotaxis, this model has developed a lot of research results. In this thesis based on the classical chemotaxis we make some expand on this model and study the qualitative properties of solutions of this extended type of models.This thesis was divided into five chapters, in the first two chapters, we present introduction and some basic knowledge, in the third and fourth chapter, we separately consider the global exist solution of the system(3.1),(4.1), in the fifth chapter, we discuss finite time blow-up solution of the system(5.1).In the third chapter of this thesis, we consider the global existence solution of parabolic-elliptic-elliptic chemotaxis model with attraction and repulsion effects, this model has more repulsion effect than the classical chemotaxis model. Firstly, we apply the Fixed Point Theorem to prove the local existence solution of model, then combined strong coupled structure and some inequalities we prove the norm, finally we use the Moser iteration to give the norm, thus we prove the model has the global existence solution by take advantage of the regularity theory.In the fourth chapter of this thesis, we discuss the global existence solution of parabolic-parabolic-parabolic chemotaxis model with attraction and repulsion effects, this model also has more repulsion effect than the classical chemotaxis model. Firstly, we also apply the Fixed Point Theorem to prove the local existence solution of model; Next we make variable change to transform a new model and find a entropy inequality, from this inequality we get a new apriori estimate,again we prove the norm, finally we also use the Moser iteration to give the norm, thus we prove the model has the global existence solution by take advantage of the regularity theory.In the fifth chapter, we study the blow up property of solution for classical chemotaxis with the nonlinear diffusion. Firstly, we the local solution; Next, under some condition about the nonlinear diffusion function and Logistic growth, we introduce two new functional and derive a differential inequality and from this we can prove the solution will blow up in finite time provided the initial data are suitably chosen.