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

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 $\theta$.