Stabilization in the Keller--Segel system with signal-dependent sensitivity

Abstract

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{align*} &u_t = Δu - χ\nabla \cdot (uS(v)\nabla v), &v_t = Δv - v + u, \end{align*} where $\chi >0$ and $S$ is a given function generalizing the sensitivity $S(s)=\frac{1}{(a+s{)}^k}$, $k>1$, $a\ge 0$, and shows exponential convergence of global classical solutions under an additional smallness condition condition for $\chi >0$.