Global Existence And Boundedness For Some Chemotaxis Models

Chemotaxis models are hot issues in the field of partial diferential equations of parabolic type.Chemotaxis refers to the directional movement of the organism along the gradient direction of its concentration under the stimulation of chemical substances.In this paper,we study the global existence and boundedness of weak or generalized solutions for several chemotaxis models.The dissertation is divided into six parts:Chapter 1 gives the biological background and research status of chemotaxis models,and briefly introduces the main work and innovations of this paper.In Chapter 2,a Keller-Segel system with nonlinear diffusion and singular sensitivity is considered under homogeneous boundary conditions of Neumann type for u and v in a bounded domain ?(?)Rn(n?2)with smooth boundary.We construct global generalized solutions of this model and show that the problem possesses a global generalized solution provided that m>1+(n-2)/(n2).In Chapter 3,we construct global generalized solutions of the initial boundary value problem where ?(?)Rn(n?1)is a bounded domain with smooth boundary,?>1 is a parameter and?,? as well as u0,v0 are sufficiently regular given functions.In particular,if ?(x)??1>0,then the condition for the existence of generalized solutions is ?>min {(2n-2)/n,(2n+4)/(n+4)}.In Chapter 4,we consider the Keller-Segel system with gradient dependent chemotactic sensitivity in a smooth bounded domain ?(?)Rn(n?2).It,is shown that the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that 1<p<n/(n-1).In Chapter 5,we deal with the Keller-Segel system with p-Laplacian diffusion and gradient dependent chemotactic sensitivity where p?2,?(?)Rn(n?2)is a bounded domain with smooth boundary.It is shown that for all reasonably regular initial data u0? 0 and v0? 0,the corresponding Neumann initialboundary value problem possesses a global weak solution which is uniformly bounded provided that q<min{(n/(n-1)(p-1)/p)+1,(p+n-2)/(n-1)}.In Chapter 6,we summarize the whole paper and put forward six problems to be studied.