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 $t\rightarrow\infty$ if $\alpha\in(0,\frac{1}{2})$ and $\mu>\mu_\star$ sufficiently large, which is moreover enjoying exponential convergence when $\alpha\in(0,\frac{n+2}{n^2+4})$.