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

This paper deals with the 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 every 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 $u(s,x;s,u_0)=u_0(x)$ in any space dimensional setting. We also proved that globally defined positive solutions of (0.1) are uniformly bounded under the assumption \begin{equation} a_{\inf}>\begin{cases}\frac{\mu \chi^2}{4} \quad {\rm if}\quad 0<\chi \leq 2\cr\mu (\chi-1)\quad {\rm if}\quad \chi>2.\end{cases}\end{equation} In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption (0.2). Among others, we provide some concrete estimates for $\int_\Omega u^{-p}$ and $\int_\Omega u^q$ for some $p>0$ and $q>2N$ 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 $t$