Global Existence And Asymptotic Behavior For The Chemotaxis Model

We consider the chemotaxis models of partial differential equations which are used in math-ematical biology to describe the spatio-temporal evolution of populations which, besides moving randomly, are able to partially direct their motion toward increasing or decresing concentrations of a chemical signal substance. The particular focus of the present dissertation is on the global solvability and asymptotic properties of some chemotaxis systems.The dissertation is organized as follows. In Chapter 2, we consider the Neumann boundary value problem for the system in a smooth bounded domain ?(?)RN (N?1), where the functions D(u) and S(u) are supposed to be smooth satisfying D(u)?Mu-? and S(u)?Mu? with M>0, ??R and ??R for all u?1, and the logistic source f(u) is smooth fulfilling f(0)? 0 as well as f(u)?a-?u? with ??0,?> 0 and ??1 for all u?0. It is shown that if then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded.Chapter 3 is concerned with the Cauchy problem for a two-species chemotactic Keller-Segel system in R2×[0,?),where ??0,x1,x2 and ?1,?2 are real numbers.It is proved that the existence of global solutions for small initial data.Moreover,we show decay properties of the small-data solution as follows:·If ?=0,the solution is asymptotic to a self-similar solution for large time.·If ?>0,the solution behaves like a multiple of the heat kernel as t??.In Chapter 4,we study an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation in a bounded convex domain ?(?)R2.It is known that if x>0,??R and ??C2(?),for suffi-ciently smooth initial data,the model possesses a unique global classical solution which satisfies (n,c,u)?(n0,0,0)as t?? uniformly with respect to x??,where n0:=1/|?| (?)? n(x,0)dx.In the present paper,we prove that this solution converges to(n0,0,0)exponentially in time.Chapter 5 deals with an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations with the porous-medium-type diffusion model in a bounded convex domain ?(?)R3.Here ??R,??W1,?(Q),0<x?C2([0,?))and 0?f?C1([0,?))with f(0)=0.It is proved that under appropriate structural assumptions on f and x,for any choice of m?2/3 and all sufficiently smooth initial data(n0,c0,u0)the model possesses at least one global weak solution.