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

This paper deals with the Keller–Segel system with signal-dependent sensitivity \begin{align*} &u_t = \Delta u - \chi \nabla \cdot (uS(v)\nabla v), &v_t = \Delta v - 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 $\chi >0$.