Studies On Properties Of Solution For Several Classes Of Nonlinear Chemotaxis Models

Chemotaxis model is a mathematical model established by partial differential equations,which is used to describe the directional movement of single or multicellular organisms toward concentration gradients of the chemical signal substance.It has a very wide range of applications of the biological field.For example,in the process of embryo differentiation,single cells form multi-cells after receiving signals to stimulate directional movement,and then form tissues and organs;When bacteria invade the body,white blood cells can deform through the capillary wall,towards the site of the invasion of bacteria,surrounding it,phagocytic.In this thesis,we consider the local existence,global existence,boundedness and blow-up of solutions of four nonlinear chemotactic models.This thesis is mainly divided into the following six chapters:In Chapter 1,we introduction the current status of the chemotaxis model,the source of the research content and the summary of the research content of this thesis.In Chapter 2,we study a parabolic-elliptic coupled chemotaxis model with nonlinear signal production and logistic source.It is proved that the solution is finite time blow-up when the logistic source and nonlinear signal production exponent in the equation meet the appropriate conditions.In Chapter 3,we discuss a quasilinear parabolic-elliptic coupled chemotaxis model.It is proved that the solution is finite time blow-up when the diffusion exponent and the nonlinear signal production exponent in the equation meet the appropriate conditions.In Chapter 4,we study the chemotaxic model of nonlinear signal with flux-limitation.By constructing the comparison function and applying the comparison principle,we obtain the local existence of the solution of the chemotaxic model;furthermore,by using the commonly used inequalities,L~p estimates and Moser-Aikakos iteration,it is obtained that the solution of the chemotaxic model is globally bounded under appropriate conditions.In Chapter 5,we investigate the attraction-repulsive chemotaxis model with rotation term.Firstly,in order to overcome the difficulties caused by the nonlinear boundary conditions and the chemotaxis sensitive function,we will deal with the boundary and the sensitive function to get the regularization problem of the model,and then give some important prior estimates to get the global bounded classical solution to the regularization problem;Secondly,the global bounded weak solution to the attraction-repulsion model with rotation term is obtained by the approximation method and the definition of the weak solution.In Chapter 6,we summarize the research works in this thesis and give the prospect of future researchs issues.