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

Source: [Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence](<https://www.sota2.com/research/paper/68fa5b971e209131ce9c9c7a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:09:39.080Z

Authors: Halil Ibrahim Kurt, Wenxian Shen

Published: 2022-12-19T20:30:02.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2212.09838>)

[PDF](<https://arxiv.org/pdf/2212.09838.pdf>)