Asymptotics in a two-species chemotaxis system with logistic source

Abstract

This paper deals with nonnegative solutions of a fully parabolic two-species chemotaxis system with competitive kinetics under homogeneous Neumann boundary conditions in a N-dimensional bounded smooth domain with reasonably smooth nonnegative initial data. In a previous paper of Bai & Winkler (2016), the equilibrium of the global bounded classical solution was shown in both coexistence and extinction cases. We extend this result to weak solutions and prove these solutions globally exist and finally converge to the same semi-trivial steady state in a certain sense.