The Existence And Asymptotic Behavior Of Solutions For Some Chemotaxis Models

The present thesis studies several chemotaxis models in biomathematics,which describe the motion of cells,who,besides random diffusion,bias their movement towards a chemically more favorable environment,including the chemotaxis-hapotaxis model and two types of chemotaxis-fluid models.This thesis is devoted to studying the well-posedness,global boundedness and the small-convection limit for these chemotaxis models.The dissertation is divided into four parts.Chapter 1 gives a summary to the chemotaxis models involving the study background and our main results.In Chapter 2,a quasi-linear chemotaxis-hapotaxis model with logistic source(?)is considered under homogeneous Neumann boundary conditions in a smooth bounded domain?(?)RN,where ?>0,?>0,?>0 and r>1.The diffusivity D(u)stasifies D(u)? CDum-1 for all u>0 with some constant CD>0.By application of the Lp esimate,Moser-Alikakos iteration mehtod and continuity argument,we have proved that the indexes in nonlinear diffusion and the logistic term,which satisfies(?)have guaranteed the global boundedness of classical solutions.This result generalize and improve some previously known ones.In Chapter 3,a chemotaxis-fluid system with nonlinear diffusion and rotational flux given by(?)is considered under omogeneous boundary conditions of Neumann type for n and c,and of Dirichlet type for u,where ?(?)R3 is bounded domain with smooth boundary condition,??W1,?(?),the tensor-valued function S(x,n,c)is dominated by |S(x,n,c)| ? S0(n+1)-?(?>0).For all sufficiently smooth initial data(n0,c0,u0),if m+?>4/3 and m>1/3,the corresponding initial-boundary value problem possesses at least one global generalized weak solution.This result generalize and improve some previously known ones..In chapter 4,a four-component chemotaxis-fluid system(?)is considered,which is used to model coral fertilization.Here,n,c,and m satisfy hemogenous Neumann boundary,u fulfills homogeneous Dirichlet boundary condition,?(?)R2 is a bounded domain with smooth bouandry.For any k ?(-1,1),assume that(nk,ck,mk,uk)is the corre-sponding global classical solution and the initial data is smooth enough,then(nk,ck,mk,uk)will stabilize to(n0,c0,m0,u0)with an explicit rate and a time dependent coefficient as k? 0.