Asymptotic Behavior of Solutions to a Logistic Chemotaxis System with Singular Sensitivity 
Received:September 07, 2020 Revised:January 03, 2021 
Key Word:
asymptotic behavior chemotaxis singular sensitivity logistic source

Fund ProjectL:Supported by Innovation Team of China West Normal University (Grant No.CXTD20205) and the Meritocracy Research Funds of China West Normal University (Grant No.17YC372). 

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Abstract: 
In this paper, we study the asymptotic behavior of solutions to the parabolicelliptic chemotaxis system with singular sensitivity and logistic source $$\left\{\begin{array}{ll}u_t=\Delta u\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta vv+u,&x\in\Omega,\,t>0\end{array}\right.$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n~(n>2)$ with the nonflux boundary, where $\chi$, $r$, $\mu>0$, $k\geq2$. It is proved that the global bounded classical solution will exponentially converge to $((\frac{r}{\mu})^{\frac{1}{k1}}, (\frac{r}{\mu})^{\frac{1}{k1}})$ as $t\rightarrow\infty$ if $r$ is suitably large. 
Citation: 
DOI:10.3770/j.issn:20952651.2021.05.004 
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