Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals

In this paper, we investigate the competitive parabolic-elliptic-parabolic-elliptic two-species chemotaxis system with two chemicals in a smooth bounded domain Ω⊂Rn (n≥1). The existence of global bounded classical solution is proved under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects. Moreover, for the case when both competition parameters a1 and a2 lie in 0,1, it is shown that such solution stabilizes to spatially homogeneous equilibria in the large time limit.