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This paper focuses on the following Keller-Segel system with singularsensitivity 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 valueproblem 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 stationarysolution $(\frac{a}{\mu }, \frac{a}{\mu })$ in the norm of $L^\infty(\Omega)$as $t\rightarrow\infty$. To the best of our knowledge, this new result is {\bfthe first} analytical work for the boundedness and {\bf asymptotic behavior} ofKeller--Segel system with {\bf singular sensitivity} and {\bf logistic source}in higher dimension case ($N\geq3$).

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