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=Δu-χ\nabla\cdot(\frac{u}{v^α}\nabla v) +ru-μu^2$, $0=Δv-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$ for $n\ge 2$. where $α\in(0,1)$ and $χ,r,μ>0$. If $α\in(0,\frac{n+2}{2n})$ and $μ>μ_0$ with $μ_0>0$ suitably large, we give the explicit expression of the upper bound for $u$ with respect to the coefficient $μ$ 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 $z=v^{1-α}$, it is proved that the solution $(u,v)$ converges to $(\frac{r}μ,\frac{r}μ)$ in $L^\infty$-norm as $t\rightarrow\infty$ if $α\in(0,\frac{1}{2})$ and $μ>μ_\star$ sufficiently large, which is moreover enjoying exponential convergence when $α\in(0,\frac{n+2}{n^2+4})$.