Properties Of Solutions For A Class Of Chemotaxis Models

In this paper,we investigate the global existence of weak solutions for the following chemotaxis model with singular sensitivity and nonlinear diffusion,(?)where Q=QxR+,? is a bounded domain of RN(N?2)with smooth boundary,m>0,x>0,?>0,0<r<4/(N?2).Here u,v represent the cell density,signal concentration respectively,the chemotactic sensitivity is represented by x/v(when sensitivity is positive,it means chemotactic attraction,and when sensitivity is negative,it means chemotactic repulsion),the consumption of the signal is given by-vur,the proliferation or death of cells according to the logistic law ?u(1-u)with ?>0.The partial differential equations coupled by two parabolic equations are a special chemotaxis model,chemotaxis model describes that cells will not only spread towards the direction of low cell concentration,but also move towards the direction of high chemotaxis signal concentration(when chemotaxis sensitivity is greater than 0)under the influence of chemotaxis signal concentration.Therefore,in nature,chemotaxis model can be used to describe both the micro world(tumor formation,cancer cell proliferation,stem cell proliferation and differentiation)and the macro world(population migration,predation,control and prevention of infectious diseases,etc).It is also of practical significance to study the properties of chemotaxis model solutions.For example,if the solutions are blow-up in a finite time,it means that cells(or biological population)will appear highly concentrated phenomenon;if the initial value is given,the study of the asymptotic behavior of the solution will be helpful to understand the density distribution of cell population after a long time,and so on.Therefore,the properties of chemotaxis model solutions attract extensive interest of author.In question(1.1),we consider a chemotaxis-consumption model with nonlinear diffusion and singular sensitivity in a bounded domain and zero-flux boundary conditions,and obtain the global existence and local boundedness of weak solutions.In the first chapter of this paper,we mainly introduce the research background and research status of chemotaxis equation.We hope to find the research methods or ideas of chemotaxis model through the research and understanding of these research results and processes.Then we summarize the structure and main content of this paper.In the second chapter,we give the definition of related symbols or nouns,and the related inequalities and related lemmas in the main result.In the third chapter,according to the definition of weak solution of general parabolic equation and Neumann initial boundary condition,the definition of weak solution of chemotaxis model in this paper is given.Next,because the problem(1.1)has singular sensitivity,it is impossible to use the fixed point theorem directly,so we give its regularization problem and give its alternative theorem.Then,a special logarithmic transformation is constructed,and the local lower bound of chemotaxis signal is obtained by using comparison principle and Neumann heat semigroup.Furthermore,the regularization estimates for slow diffusion {m>1)and fast diffusion(0<m?1)are discussed respectively.Finally,the global existence and local boundedness of weak solutions for the chemotaxis model are obtained.The fourth chapter is the conclusion and Prospect of this paper.The research methods and results of this paper are summarized,as well as the prospect of its classical solution.