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.CXTD2020-5) and the Meritocracy Research Funds of China West Normal University (Grant No.17YC372).
 Author Name Affiliation Wanjuan DU Institute for Advanced Studies in the History of Science, Northwest University, Shaanxi 710217, P. R. China College of Mathematics Education, China West Normal University, Sichuan 637009, P. R. China
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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 $$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n~(n>2)$ 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 $r$ is suitably large.