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

TL;DR

This paper demonstrates that strong logistic growth terms can prevent finite-time blow-up in chemotaxis models with singular sensitivity across any spatial dimension, ensuring global existence and boundedness of solutions.

Contribution

It proves global existence and boundedness of solutions in chemotaxis models with singular sensitivity using logistic kinetics, extending results to any spatial dimension.

Findings

Logistic source enforces global solutions in all dimensions.

Solutions remain bounded under specific conditions on growth rates.

Finite-time blow-up is prevented by logistic kinetics even with large chemotactic sensitivity.

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.