Global solvability and asymptotic behavior in a two‐species chemotaxis system with two chemicals

Abstract

This paper deals with a two‐species chemotaxis system with Lotka–Volterra competitive kinetics under homogeneous Neumann boundary condition in a smooth domain , where the parameters , and are positive constants for . Under appropriate regularity assumptions on the initial data, we obtain that the system possesses a globally bounded weak solution. Furthermore, we establish the asymptotic stabilization of this weak solution by constructing suitable energy functions in both coexistence and extinction cases.