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 $\Omega \subset {\mathbb{R}}^N(N\ge 1)$, 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 $t\to \infty$. 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 ($N\ge 3$).

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References in corpus (2)

• Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions
• A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data