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, \begin{equation} \begin{cases} u_t=\Delta u-\chi_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in \Omega\cr v_t=\Delta v-\chi_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in \Omega\cr 0=\Delta w-\mu w +\nu u+ \lambda v,\quad &x\in \Omega \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partial\Omega, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, and $\chi_i,a_i, b_i, c_i$ ($i=1,2$) and $\mu,\, \nu, \, \lambda$ are positive constants. In [25], we proved that for any given nonnegative initial data $u_0,v_0\in C^0(\bar\Omega)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution provided that $\min\{a_1,a_2\}$ is large relative to $\chi_1,\chi_2$, and $u_0+v_0$ is not small in the case that $(\chi_1-\chi_2)^2\le \max\{4\chi_1,4\chi_2\}$ and $u_0+v_0$ is neither small nor big in the case that $(\chi_1-\chi_2)^2>\max\{4\chi_1,4\chi_2\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^*,v^*,w^*)$, where $$u^*=\frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2},\quad v^*=\frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^*=\frac{\nu}{\mu}u^*+\frac{\lambda}{\mu} v^*.$$ We obtain some explicit conditions on $\chi_1,\chi_2$ which ensure that the positive constant solution $(u^*,v^*,w^*)$ is globally stable in the sense that for any given nonnegative initial data $u_0,v_0\in C^0(\bar\Omega)$ with $u_0\not \equiv 0$ and $v_0\not \equiv 0$, $$\lim_{t\to\infty}\Big(\|u(t,\cdot;u_0,v_0)-u^*\|_\infty +\|v(t,\cdot;u_0,v_0)-v^*\|_\infty+\|w(t,\cdot;u_0,v_0)-w^*\|_\infty\Big)=0.$$