Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

TL;DR

This paper studies a chemotaxis system with singular sensitivity and logistic growth, establishing boundedness, persistence, invariant sets, and existence of entire solutions, advancing understanding of long-term behavior of such biological models.

Contribution

It provides new bounds, invariant sets, and existence results for positive solutions of the chemotaxis system with singular sensitivity and logistic source.

Findings

Globally bounded positive solutions are established.

Existence of a bounded invariant set attracting all solutions.

Existence of positive entire classical solutions, periodic or steady-state.

Abstract

This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(t,x)-b(t,x) u), & x\in \Omega,\cr 0=\Delta v- \mu v+ \nu u, & x\in \Omega, \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partial\Omega, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $a(t,x)$ and $b(t,x)$ are positive smooth functions, and $\chi$, $\mu$ and $\nu$ are positive constants. In the very recent paper [25], we proved that for given nonnegative initial function $0\not\equiv u_0\in C^0(\bar \Omega)$ and $s\in\mathbb{R}$, (0.1) has a unique globally defined classical solution $(u(t,x;s,u_0),v(t,x;s,u_0))$ with $u(s,x;s,u_0)=u_0(x)$, provided that $a_{\inf}=\inf_{t\in\mathbb{R},x\in\Omega}a(t,x)$ is large relative to $\chi$ and $u_0$ is not small. In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption that $a_{\inf}$ is large relative to $\chi$ and $u_0$ is not small. Among others, we provide some concrete estimates for $\int_\Omega u^{-p}$ and $\int_\Omega u^q$ for some $p>0$ and $q>\max\{2,N\}$ and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a ``rectangular'' type bounded invariant set (in $L^q$) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution $(u^*(t,x),v^*(t,x))$, which is periodic in $t$ if $a(t,x)$ and $b(t,x)$ are periodic in $t$ and is independent of $t$ if $a(t,x)$ and $b(t,x)$ are independent of $t$.