Boundedness And Asymptotic Behavior Of Solutions For Multi-species And Nonlinear Single-species Chemotaxis Models

Chemotaxis models are used to describe the directional movement of microorganisms or biological cells when stimulated by certain chemical signaling substances.Chemotaxis models have been widely used,such as cellular immune response,wound healing,and microbial proliferation.In this thesis,the properties of solutions of two kinds of chemotaxis models are studied: one is the multi-species chemotaxis model,in linear and nonlinear forms,the global boundedness and long time behavior of solutions are studied;the other class is the nonlinear single-species chemotaxis model without Logistic source and the global boundedness of its solution is studied.Specific as follows:Firstly,the linear chemotaxis model of two-species and two-chemical is studied in Chapter 2,which depicts the phenomenon of competition between two species for multiplication.In this chapter,it is studied that the solutions of model is global boundedness when both growth coefficients are sufficiently large by using a prior estimates and undetermined coefficient method in three-dimensional space.Secondly,on the basis of the model in Chapter 2,the nonlinear sensitivity terms,Logistic sources terms and signal secretion terms are introduced in Chapter 3.By constructing appropriate energy functional inequality and using undetermined coefficient method,the global boundedness of solution to the model is obtained under14 different parameters cases,which partially improves the results about the parameter range in reference [74,Theorem 3.1].Then,on the basis of the first two chapters,a more realistic model is considered in the Chapter 4,the nonlinear diffusion terms are introduced,that is to say,the species no longer diffuses only in a linear form.By constructing different prior estimates,the boundedness of solutions of the model in the absence of Logistic source and with Logistic source are obtained,respectively;the long-term behavior of the corresponding bounded solution with Logistic source is obtained by constructing different Lyapunov functionals in the case of two parameters cases.Finally,in Chapter 5,a nonlinear single-species chemotaxis model with nonlinear signal secretion is studied.By constructing an appropriate energy functional inequality,the global boundedness of the model solution is obtained.This result removes the assumption on the upper bound of the diffusion function in the previous reference [109,Theorem 1.1] and gives the necessary limitation on the diffusion function index and the sensitivity function index.