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

TL;DR

It is shown that, in any space dimensional setting, 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 $\chi$.

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=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+σ}),\quad &x\in Ω\cr 0=Δv-μv+νu,\quad &x\in Ω\quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partialΩ$, $χ$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $μ,ν$ are positive constants, and $σ\ge 0$. When $σ>0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar Ω)$, (0.1) has a unique globally defined classical solution $(u_σ(x,t;u_0),v_σ(x,t;u_0))$ with $u_σ(x,0;u_0)=u_0(x)$, 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 $χ$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{μχ^2}{4}, &\text{if $0< χ\leq 2,$}\\ μ(χ-1), &\text{if $χ>2$.}\\ \end{cases} \end{equation*} When $σ=0$, 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.

Theorems & Definitions (44)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof