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

TL;DR

This paper proves exponential convergence of global classical solutions in a Keller--Segel system with signal-dependent sensitivity, under a smallness condition on the chemotactic sensitivity parameter.

Contribution

It establishes the stabilization and exponential convergence of solutions for a generalized Keller--Segel model with signal-dependent sensitivity, extending previous results.

Findings

Global classical solutions converge exponentially.

Stability is achieved under a smallness condition on .

Generalizes sensitivity functions beyond the standard form.

Abstract

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 $\chi>0$.