Stabilization in a two-species chemotaxis system with two chemicals under small initial conditions

Abstract

This paper deals the two-species chemotaxis system in a smoothly bounded domain Ω⊂Rn(n≥1). It is proved that the solution of the system is globally bounded and exponentially convergent to the constant steady state solution under small initial conditions ‖u0‖Ln+β(Ω) and ‖ω0‖Ln+β(Ω) for any β>0, and the convergence rates are calculated accurately. The present work improves the results under small initial conditions ‖u0‖L∞(Ω) and ‖ω0‖L∞(Ω) in Tao and Winkler (2015).