Global existence, boundedness and asymptotic behavior of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity

Abstract

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics(0.1){ut=Δu−χ1∇⋅(uw∇w)+μ1u(1−u−a1v),t>0,x∈Ω,vt=Δv−χ2∇⋅(vw∇w)+μ2v(1−v−a2u),t>0,x∈Ω,wt=Δw−w+u+v,t>0,x∈Ω,∂u∂ν=∂v∂ν=∂w∂ν=0,t>0,x∈∂Ω,u(0,x)=u0(x),v(0,x)=v0(x),w(0,x)=w0(x),x∈Ω, where Ω⊂RN(N≥2) is a bounded smooth convex domain, and the parameters χ1,χ2,μ1,μ2,a1 and a2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ1,χ2∈(0,12)forN=2,3 or χ1,χ2∈(0,N−2N−1)forN≥4. Moreover, if min⁡{μ1,μ2}>max⁡{χ1,χ2}24, we obtain the uniformly lower bound for w. Finally, when χ1,χ2 are suitably small, it is proved that if 0<a1,a2<1, then the solution (u,v,w) converges to (1−a11−a1a2,1−a21−a1a2,2−a1−a21−a1a2) in L∞ norm as t→∞; if 0<a2<1≤a1, then the solution (u,v,w) converges to (0,1,1) in L∞ norm as t→∞.