In this paper, we study the following chemotaxis system with singular sensitivity and logistic source

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{u}_t=\mathrm{\Delta }u-\chi \nabla ·\left({\displaystyle \frac{u}{v}\nabla v}\right)+ru-\mu u^k,\hfill & x\in \mathrm{\Omega },\hfill & t>0,\hfill \\ v_t=\mathrm{\Delta }v-v+u,\hfill & x\in \mathrm{\Omega },\hfill & t>0,\hfill \end{array}\right.\end{array}$$

in a smooth bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( $n\ge 2$) with the non-flux boundary, where $\chi ,r,\mu >0,k>2$. The boundedness of solutions has been proved in the case that $n\ge 2$, $k>\frac{3(n+2)}{n+4}$ and r, $\chi >0$ satisfying ${\chi }^2<\min \left\{\frac{2\left(r+r^2\right)}{k},\frac{4}{k(k-1)(k-2)}\right\}$ (Zhao and Zheng, 2019). This paper mainly aims to give the large time behavior of bounded solutions and prove that the global classical solution will exponentially converge to $\left({\left(\frac{r}{\mu }\right)}^{1/(k-1)},{\left(\frac{r}{\mu }\right)}^{1/(k-1)}\right)$ as $t\to \infty$ if μ is suitably large.

中文翻译：

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具有单一敏感性和逻辑源的趋化系统解的长时间行为

在本文中，我们研究了以下具有奇异敏感性和逻辑源的趋化系统

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{你}_{吨}=\mathrm{\Delta }你-\chi \nabla ·\left({\displaystyle \frac{你}{v}\nabla v}\right)+r你-\mu {你}^{克},\hfill & \times \in \mathrm{\Omega },\hfill & 吨>0,\hfill \\ v_{吨}=\mathrm{\Delta }v-v+你,\hfill & \times \in \mathrm{\Omega },\hfill & 吨>0,\hfill \end{array}\right.\end{array}$$

在光滑有界凸域中 $\mathrm{\Omega }\subset {\mathbb{电阻}}^n$ ( $n\ge 2$) 与非通量边界，其中 $\chi ,r,\mu >0,克>2$. 解的有界性已在以下情况下得到证明$n\ge 2$, $克>\frac{3(n+2)}{n+4}$和[R ，$\chi >0$ 满意 ${\chi }^2<\mathrm{分钟}\left\{\frac{2\left(r+r^2\right)}{克},\frac{4}{克(克-1)(克-2)}\right\}$（赵和郑，2019）。本文主要旨在给出有界解的大时间行为，并证明全局经典解将指数收敛到$\left({\left(\frac{r}{\mu }\right)}^{1/(克-1)},{\left(\frac{r}{\mu }\right)}^{1/(克-1)}\right)$ 作为 $吨\to \infty$ 如果 μ 适当大。