Qualitative Study Of Two-species And Two-stimuli Chemotaxis Models

In this paper,the properties of solutions to chemotaxis models with two-species and two-stimuli are studied.Firstly,the initial boundary value problem to the following two-species and two-stimuli chemotaxis model with linear chemotactic sensitivity is studied:Here ?(?)R2 are bounded domains with smooth boundary,(?)denotes the derivative with respect to the outward normal on(?)?.By using the fixed point theory,a series of energy estimates and an entropy functional inequality,the global existence and uniform boundedness of the classical solution with small initial data are established.Secondly,the two-dimensional homogeneous Neumann initial boundary value problem of the following two-species and two-stimuli chemotaxis model with nonlinear chemotactic sensitivity is studied:Here the chemotactic sensitivity functions f,g ?C1-?([0,?))(?>0)are given functions satisfying f(0)=0,g(0)=0 and with sublinear growth.In this paper,we first establish a series of a prior energy estimates of the solution to this initial boundary value problem,based on these energy estimates,the boundedness of the entropies are obtained,and then the global existence and boundedness of solution with large initial data are proved.