Boundedness and large time behavior in a higher-dimensional Keller--Segel system with singular sensitivity and logistic source

This paper focuses on the following Keller-Segel system with singular sensitivity and logistic source $$\left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{ v_t=\Delta v- v+u},\quad x\in \Omega, t>0 \end{array}\right.\eqno(\star)$$ in a smoothly bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with zero-flux boundary conditions, where $a>0,\mu>0$ and $\chi>0$ are given constants. If $\chi$ is small enough, then, for all reasonable regular initial data, a corresponding initial-boundary value problem for $(\star)$ possesses a global classical solution $(u, v)$ which is {\bf bounded} in $\Omega\times(0,+\infty)$. Moreover, if $\mu$ is large enough, the solution $(u, v)$ exponentially converges to the constant stationary solution $(\frac{a}{\mu }, \frac{a}{\mu })$ in the norm of $L^\infty(\Omega)$ as $t\rightarrow\infty$. To the best of our knowledge, this new result is {\bf the first} analytical work for the boundedness and {\bf asymptotic behavior} of Keller--Segel system with {\bf singular sensitivity} and {\bf logistic source} in higher dimension case ($N\geq3$).