Qualitative Analysis Of Solutions To Some Chemotaxis Models

This thesis studies several chemotaxis models in biomathematics,which describe the biased movement of cells in response to the concentration gradient of chemical signal.Our models include chemotaxis model with consumption,prey-taxis model and forager-exploiter model.This thesis is devoted to studying the global existence and asymptotic behavior of these models.The dissertation is divided into five parts.Chapter 1 gives an overview to the chemotaxis models involving the study background and our main results.In Chapter 2,we investigate a high-dimensional chemotaxis system with consumption of chemo attract ant (?) under homogeneous boundary conditions of Neumann type,in a bounded domain ??Rn(n?4)with smooth boundary.It is proved that if initial data satisfy u0?Co(?)and v0?W1,q(?)for some q>n,the corresponding initial-boundary value problem possesses at least one global renormalized solution.In Chapter 3,we consider a parabolic-parabolic system with gradient dependent chemotactic coefficient and consumption of chemoattractant under homogeneous boundary conditions of Neumann type,in a bounded domain ??Rn(n?2)with smooth boundary,1<p<2.It is proved that if initial data satisfy u0 ? C0(?),(?)for some q>n and 0<?v0?L?(?)<1/(4K),then the model admits at least one global weak solution for n<8-2(p-1)/p-1 and possesses at least one global renormalized solution for n?8-2(p-1)/p-1.Here,(?)is positive and finite.In chapter 4,we deal with a prey-taxis system with rotational flux terms under no-flux boundary conditions in a bounded domain ? ?Rn(n? 1)with smooth boundary.Here the matrix-valued function S E C2(?×[0,?)2;Rn ×n)fulfills |S(x,u,v)|?S0(v)/(1+u)?(??0)for all(x,u,v)??×[0,?)2 with some nondecreasing function S0.It is proved that for nonnegative initial data uo?Co(?)andtv0 ? W1,q(?)with some q>max{n,2},if one of the following assumptions holds:(?)n=1,(?)n>2,?=0?(?)(?)?>0,then the model possesses a global classical solution that is uniformly bounded.Where m:=max{?v0?L?(?),1/??.In Chapter 5,we study forager-exploiter model with super linear degradation under homogeneous boundary conditions of Neumann type,in a bounded domain ??Rn(n?2)with smooth boundary.Here ?,?>0,r?C1(?×[0,?))?L?(?×(0,?))is nonnegative and the functions f,g?Cl([0,?))fulfill f(0)? 0,g(0)? 0 as well as-kfs?-lf?f(s)?-Kfs?+Lf and-kgs?-lg?g(s)??Kgs?+Lg for s?0,respectively,with constants ?,?>1,kf,Kf,kg,Kg>0 and lf,Lf,lg,Lg?0.It is proved that?,?>?(n+2)/n and ?>1-(2/n+2)min{?,?},the model possesses a global classical solution that is uniformly bounded.Moreover,stabilization of these solutions toward homogeneous equilibria is established in the large time limit,provided that the integration ?1???r2 is finite.