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=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i$, $a_i$, $b_i$, $ c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ 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 $u_0,v_0\in C^0(\barΩ)$ 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 $χ_1,χ_2$ 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\inΩ}(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.

Theorems & Definitions (53)

• Definition 1.1
• Proposition 1.1: Local existence
• Theorem 1.1: Local lower bound of the combined mass
• Remark 1.1
• Theorem 1.2: Local $L^p$- and $C^\theta$-boundedness
• Remark 1.2
• Theorem 1.3: Global existence
• Remark 1.3
• Theorem 1.4
• Remark 1.4