Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion

Abstract

This paper deals with the two-species chemotaxis system{ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u)inΩ×(0,∞),vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v)inΩ×(0,∞),wt=dΔw+h(u,v,w)inΩ×(0,∞), where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n∈N; h, χi are functions satisfying some conditions. Negreanu–Tello (2014, 2015) [12,13] proved global existence and asymptotic behavior of solutions to the above system when 0≤d<1. The main purpose of the present paper is to remove the restriction of 0≤d<1.