Boundedness and stabilization in a two-species chemotaxis system with signal absorption

This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=\Delta u-\chi_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*} v_t=\Delta v-\chi_2\nabla\cdot(v\nabla w), \end{equation*} \begin{equation*} w_t=\Delta w-(\alpha u+\beta v)w \end{equation*} in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ ($n\geq2$), where the parameters $\chi_1$, $\chi_2$, $\alpha$ and $\beta$ are positive. It is proved that if \begin{equation*} \max\{\chi_1,\chi_2\}\|w(x,0)\|_{L^{\infty}(\Omega)}<\sqrt{\frac{2}{n}}\pi \end{equation*} the problem possesses a unique global classical solution that is uniformly bounded. Moreover, we prove that \begin{equation*} u(x,t)\to\frac{1}{|\Omega|}\int_{\Omega}u(x,0),\quad v(x,t)\to\frac{1}{|\Omega|}\int_{\Omega}v(x,0)\quad\mbox{and}\quad w(x,t)\to0\quad\mbox{as}\ t\to\infty \end{equation*} uniformly with respect $x\in\Omega$.