Long-time behavior of solution to a chemotaxis system with weakly singular sensitivity and logistic source

Abstract

This paper is concerned with the parabolic-elliptic chemotaxis system with weakly singular sensitivity and logistic source:~$u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v^{\alpha }}\nabla v)+ru-\mu u^2$, $0=\Delta v-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $\Omega \subset {\mathbb{R}}^n$ for $n\ge 2$. where $\alpha \in (0,1)$ and $\chi ,r,\mu >0$. If $\alpha \in (0,\frac{n+2}{2n})$ and $\mu >{\mu }_0$ with ${\mu }_0>0$ suitably large, we give the explicit expression of the upper bound for $u$ with respect to the coefficient $\mu$ after some time, without establishing the uniformly positive bound for $v$ from below. Furthermore, by dealing with the corresponding non-singular chemotaxis system via the transformation $z=v^{1-\alpha }$, it is proved that the solution $(u,v)$ converges to $(\frac{r}{\mu },\frac{r}{\mu })$ in $L^{\infty }$-norm as $t\to \infty$ if $\alpha \in (0,\frac{1}{2})$ and $\mu >{\mu }_{\star }$ sufficiently large, which is moreover enjoying exponential convergence when $\alpha \in (0,\frac{n+2}{n^2+4})$.