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

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 $t$.