Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, {equation} {cases} u_t=Δ u-χ_1 ∇· (u/w ∇ w)+u(a_1-b_1u-c_1v) , &x∈ Ω v_t=Δ v-χ_2 ∇· (v/w ∇ w)+v(a_2-b_2v-c_2u), &x∈ Ω 0=Δ w-μ w +ν u+ λ v, &x∈ Ω ∂ u/∂ n=∂ v/∂ n=∂ w/∂ n=0, &x∈∂Ω, {cases} {equation} where Ω ⊂ RN is a bounded smooth domain, and χᵢ,aᵢ, bᵢ, cᵢ ($i=1,2$) and μ,\, ν, \, λ are positive constants. In [25], we proved that for any given nonnegative initial data u₀,v₀∈ C⁰(Ω) with u₀+v₀ ≡ 0, (0.1) has a unique globally defined classical solution provided that min₁,a₂\ is large relative to χ₁,χ₂, and u₀+v₀ is not small in the case that (χ₁-χ₂)²≤ max\4χ₁,4χ₂\ and u₀+v₀ is neither small nor big in the case that (χ₁-χ₂)²>max\4χ₁,4χ₂\. In this paper, we proved that (0.1) has a unique positive constant solution (u^*,v^*,w^*), where u^*=a₁b₂-c₁a₂/b₁b₂-c₁c₂, v^*=b₁a₂-a₁c₂/b₁b₂-c₁c₂, w^*=ν/^*+λ/μ v^*. We obtain some explicit conditions on χ₁,χ₂ which ensure that the positive constant solution (u^*,v^*,w^*) is globally stable in the sense that for any given nonnegative initial data u₀,v₀∈ C⁰(Ω) with u₀ ≡ 0 and v₀ ≡ 0, limt→∞(\|u(t,·;u₀,v₀)-u^*\|_∞ +\|v(t,·;u₀,v₀)-v^*\|_∞+\|w(t,·;u₀,v₀)-w^*\|_∞)=0.