Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

Abstract

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant{ut=d1Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=d2Δv−∇⋅(vχ2(w)∇w)+μ2v(1−a2u−v),x∈Ω,t>0,wt=d3Δw−(αu+βv)w,x∈Ω,t>0under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥1) with smooth boundary, where the initial data (u0,v0)∈(C0(Ω‾))2 and w0∈W1,∞(Ω) are non-negative and the parameters d1,d2,d3>0, μ1,μ2>0, a1,a2>0 and α,β>0. The chemotactic function χi(w) (i=1,2) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for i=1,2,(i) χi(w)=χ0,i>0 and‖w0‖L∞(Ω)<πdid3n+1χ0,i−2did3n+1χ0,iarctan⁡di−d32n+1did3;(ii) 0<‖w0‖L∞(Ω)≤d33(n+1)‖χi‖L∞[0,‖w0‖L∞(Ω)]min⁡{2didi+d3,1}.Moreover, we prove asymptotic stabilization of solutions in the sense that:• If a1,a2∈(0,1) and u0≠0≠v0, then any global bounded solution exponentially converge to (1−a11−a1a2,1−a21−a1a2,0) as t→∞;• If a1>1>a2>0 and v0≠0, then any global bounded solution exponentially converge to (0,1,0) as t→∞;• If a1=1>a2>0 and v0≠0, then any global bounded solution algebraically converge to (0,1,0) as t→∞.