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

This paper deals with the homogeneous Neumann boundary-value problem for the Keller–Segel system \begin{align*} \begin{cases} u_t=Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v),\[] v_t=Δv - v + u^θ \end{cases} \end{align*} in $n$-dimensional bounded smooth domains for suitably regular nonnegative initial data, where $\chi >0$, $p\in (1,\mathrm{\infty })$ 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 $\theta$.