Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

Abstract

This paper is concerned with 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. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove 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 $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ provided that $\min\{a_1,a_2\}$ is large relative to $\chi_1,\chi_2$ and $u_0+v_0$ is not small. Moreover, under the same condition, we prove that \begin{equation*} \limsup_{t\to\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\le M^*, \end{equation*} and \begin{equation*} \liminf_{t\to\infty} \inf_{x\in\Omega}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*, \end{equation*} for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.