Stability of constant equilibria in a Keller--Segel system with gradient dependent chemotactic sensitivity and sublinear signal production

TL;DR

This paper investigates the stability of constant equilibria in a Keller--Segel chemotaxis model with gradient-dependent sensitivity and sublinear signal production, extending previous results to a broader parameter range.

Contribution

It generalizes existing stability results for the Keller--Segel system to include sublinear signal production with variable chemotactic sensitivity.

Findings

Proves stability of homogeneous equilibria under certain conditions.

Extends prior work from linear to sublinear signal production.

Provides conditions on parameters for stability.

Abstract

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=\Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=\Delta v - v + u^{\theta} \end{cases} \end{align*} in $n$-dimensional bounded smooth domains for suitably regular nonnegative initial data, where $\chi > 0$, $p \in (1, \infty)$ and $\theta \in (0,1]$. Under smallness conditions on $p$ and $\theta$, we prove that the spatially homogeneous equilibrium solution is stable. This generalizes the result in Kohatsu--Yokota (Le Matematiche, 2023; 78; 213--237) from the case $\theta = 1$ to more general values of $\theta$.