# Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

Source: [Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting](<https://www.sota2.com/research/paper/68fa5b781e209131ce9c8832>)

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

Data updated: 2025-10-25T17:04:36.402Z

Authors: Halil Ibrahim Kurt, Wenxian Shen

Published: 2020-08-05T00:23:48.000Z

## Abstract

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \\begin{equation} \\begin{cases} u\_t=\\Delta u-\\chi\\nabla\\cdot (\\frac{u}{v} \\nabla v)+u(a(x,t)-b(x,t) u^{1+\\sigma}),\\quad &x\\in \\Omega\\cr 0=\\Delta v-\\mu v+\\nu u,\\quad &x\\in \\Omega \\quad \\cr\\frac{\\partial u}{\\partial n}=\\frac{\\partial v}{\\partial n}=0,\\quad &x\\in\\partial\\Omega, \\end{cases} \\end{equation} where $\\Omega \\subset \\mathbb{R}^n$ is a bounded domain with smooth boundary $\\partial\\Omega$, $\\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $\\mu,\\nu$ are positive constants, and $\\sigma\\ge 0$. When $\\sigma\>0$, we prove that, for every given nonnegative initial data $0\\not\\equiv u\_0\\in C^0(\\bar \\Omega)$, (0.1) has a unique globally defined classical solution $(u\_\\sigma(x,t;u\_0),v\_\\sigma(x,t;u\_0))$ with $u\_\\sigma(x,0;u\_0)=u\_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions \\begin{equation\*}   a\_{\\inf}\> \\begin{cases} \\frac{\\mu \\chi^2}{4}, &\\text{if $0\< \\chi \\leq 2,$}\\\\ \\mu(\\chi-1), &\\text{if $\\chi\>2$.}\\\\ \\end{cases} \\end{equation\*} When $\\sigma=0$, we show that the classical solution $(u(x,t;u\_0,0),v(x,t;u\_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u\_0(x)$ are not small.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2008.01887>)

[PDF](<https://arxiv.org/pdf/2008.01887.pdf>)