Qualitative Analysis Of Solutions For Several Kinds Of Chemotaxis Models With Indirect Signal Production (Consumption)

Recently,many mathematicians have begun to use the theories and methods of nonlinear partial differential equation to explain and predict some complex biological phenomena,which has played a significant role in promoting the development of biology and mathematics.Under this environment,the research object of this thesis is the Keller-Segel chemotaxis model with certain biological backgrounds.Chemotaxis,refers to a directional movement of cells towards the concentration gradient of chemical signaling substances produced or consumed by the cells themselves.However,when the chemical signals are not produced or consumed by the organisms themselves,the properties of the solutions will change accordingly.This thesis makes some qualitative analysis on the chemotactic model with this property.The specific contents are as follows:After explaining the research background and content of this kind of model,first of all,the thesis considers the quasilinear chemotaxis model with indirect signal consumption.Resulting from using some basic inequalities,such as Young's inequality,Gagliardo-Nirenberg inequality,heat semi-group estimation,etc.,the global existence of the solution of this model is obtained under the condition of certain parameters.Further step,the global boundedness of solutions is derived by limiting the initial value.Finally,when the time tends to infinity,the stability of the solutions is yielded by the Poincaréinequality and some preliminaries.Then,the thesis studies the quasilinear chemotaxis-growth model with indirect signal consumption,aiming to consider the influence of growth item logistic source on the properties of the solutions based on the former model.The global boundedness of the solutions is proved in line with a series of prior estimates.Compared with the model studied before,the same boundedness conclusion can be obtained and not need limit the initial value.On the basic of the bounedness of solutions,the stable state of the solutions is implied via constructing a proper Lyapunov functional in the linear case.In the end,the thesis analyzes the properties of solutions of the singular chemotaxis-(growth)model with indirect signal production or consumption.Firstly,the singular chemotaxis model with indirect signal production is presented.The singular term is treated by taking advantage of the lower bound of v,then,the global boundedness and the asymptotic behavior of the solutions are obtained by applying the heat semi-group estimation,energy functional and some other methods.Secondly,the singular chemotaxis-growth model with indirect signal production is proposed.And the influence of the singular term is eliminate by means of a different method,then the global existence of the solutions is obtained via heat semi-group estimation.Finally,the singular chemotaxis-growth model with indirect signal consumption is considered,and the lower bound of v is gain through a special transformation,and then its global existence of solutions is derived by analogy.