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 $x\in\Omega$.
Categories: math.AP