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}~~~~(1) $ 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], among others, 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 $, (1) has a unique globally defined classical solution $ (u(t, x;u_0, v_0), v(t, x;u_0, v_0), w(t, x;u_0, v_0)) $ with $ u(0, x;u_0, v_0) = u_0(x) $ and $ v(0, x;u_0, v_0) = v_0(x) $ in any space dimensional setting with any positive constants $ \chi_i, a_i, b_i, c_i $ ($ i = 1, 2 $) and $ \mu, \nu, \lambda $. In this paper, we assume that the competition in (1) is weak in the sense that $ \frac{c_1}{b_2}<\frac{a_1}{a_2}, \quad \frac{c_2}{b_1}<\frac{a_2}{a_1}. $ Then (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, that is, 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\limits_{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. $