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

Abstract

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=Δu-χ_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*} v_t=Δv-χ_2\nabla\cdot(v\nabla w), \end{equation*} \begin{equation*} w_t=Δw-(αu+βv)w \end{equation*} in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$), where the parameters ${\chi }_1$, ${\chi }_2$, $\alpha$ and $\beta$ are positive. It is proved that if \begin{equation*} \max\{χ_1,χ_2\}\|w(x,0)\|_{L^{\infty}(Ω)}<\sqrt{\frac{2}{n}}π\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}{|Ω|}\int_Ωu(x,0),\quad v(x,t)\to\frac{1}{|Ω|}\int_Ωv(x,0)\quad\mbox{and}\quad w(x,t)\to0\quad\mbox{as}\ t\to\infty \end{equation*} uniformly with respect $x\in \Omega$.

12 pages