Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Abstract

Concerned is a two-species chemotaxis system with cross-diffusion and logistic source in a smooth bounded domain. When the system is parabolic-elliptic-parabolic-elliptic, if the logistic damping effect is strong enough then the solutions of the system are global, bounded and exponentially convergent to the constant steady-state solution. This is also true for the fully parabolic system. It is also shown that the damping effect is conducive to the global existence of solutions, which prevents the occurrence of blow-up in the presence of chemotaxis.