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

Abstract

This paper focuses on the following Keller-Segel system with singular sensitivity and logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ au-μu^2,\quad x\in Ω, t>0, \disp{ v_t=Δv- v+u},\quad x\in Ω, t>0 \end{array}\right.\eqno(\star) $$ in a smoothly bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with zero-flux boundary conditions, where $a>0,μ>0$ and $χ>0$ are given constants. If $χ$ 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 $Ω\times(0,+\infty)$. Moreover, if $μ$ is large enough, the solution $(u, v)$ exponentially converges to the constant stationary solution $(\frac{a}{μ}, \frac{a}{μ})$ in the norm of $L^\infty(Ω)$ 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 ($N\geq3$).