Asymptotic behavior of solutions to a logistic chemotaxis system with singular sensitivity
Received:September 07, 2020  Revised:December 25, 2020
Key Word: Asymptotic behavior   Chemotaxis   Singular sensitivity   Logistic source
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 Author Name Affiliation Address Wanjuan Du China West Normal University China West Normal University
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In this paper, we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source \begin{equation*} \left\{ \begin{aligned} &u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&\qquad x\in\Omega,\,t>0,\& 0=\Delta v-v+u,&\qquad x\in\Omega,\,t>0 \end{aligned} \right. \end{equation*} in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ ($n\ge2$) with the non-flux boundary, where $\chi,\,r,\,\mu>0,\,k\geq2$. It is proved that the global bounded classical solution will exponentially converge to $((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$ as $t\rightarrow\infty$ if $\mu$ is suitably large.