Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

This paper deals with the two-species chemotaxis-competition system

$\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} \ Ω × (0, ∞), \\ v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} \ Ω × (0, ∞), \\ w_t = d_3Δ w + α u + β v - γ w&{\rm in} \ Ω × (0, ∞), \end{cases} \end{equation*}$

where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n≥ 2$; $χ_i$ are functions satisfying some conditions. About this problem, Bai-Winkler [1] first obtained asymptotic stability in (1) under some conditions in the case that $a_1, a_2∈ (0, 1)$. Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that $a_1, a_2∈ (0, 1)$.