Large time behavior of solution to a parabolic-elliptic chemotaxis system with weak singular sensitivity and logistic kinetics: Boundedness, persistence, stability

This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source \begin{equation} \label{abstract-eq} \begin{cases} u_t=\Delta u-\chi \nabla\cdot (\frac{u}{v^{\lambda}} \nabla v) +ru- \mu u^2, \quad &x\in \Omega,\cr 0=\Delta v- \alpha v +\beta u,\quad &x\in \Omega, \cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0,\quad &x\in\partial\Omega, \end{cases}\, \end{equation} where $\Omega \subset \mathbb{R}^N (N \ge 2)$ is a smooth bounded domain, the parameters $\chi,\, r, \, \mu, \, \alpha,\,\beta$ are positive constants and $\lambda \in (0,1).$ In this article, for all suitably smooth initial data $u_0\in C^0(\bar\Omega)$ with $u_0 \not \equiv 0,$ it has been proven that: First, there exists $\mu > \mu_1^*(p,\lambda,\chi,\beta)$ such that any globally defined positive solution is $L^p(\Omega)$-bounded with $p \ge 2.$ Next, there exists $\mu > \mu_2^*(N,\lambda,\chi,\beta)$ such that any globally defined classical solutions is globally bounded. Third, there exists $\mu > \mu_3^*(N,\lambda,\chi,\beta)$ such that any globally defined positive solution is uniformly bounded above and below eventually by some positive constants that are independent of its initial function $u_0.$ Last, there exists $\mu > \mu_4^*(N,\lambda,\chi,\alpha,\beta,r,\Omega)$ such that any globally bounded classical solution to system (0.1) exponentially converges to the constant steady state $(\frac{r}{\mu},\frac{\beta}{\alpha}\frac{r}{\mu}).$