No pattern formation in a quasilinear chemotaxis model with local sensing

Abstract

Convergence to spatially homogeneous steady states is shown for a chemotaxis model with local sensing and possibly nonlinear diffusion when the intrinsic diffusion rate $\varphi$ dominates the inverse of the chemotactic motility function $\gamma$, in the sense that $(\varphi \gamma {)}^{\prime }\ge 0$. This result encompasses and complies with the analysis and numerical simulations performed in Choi \& Kim (2023). The proof involves two steps: first, a Liapunov functional is constructed when $\varphi \gamma$ is non-decreasing. The convergence proof relies on a detailed study of the dissipation of the Liapunov functional and requires additional technical assumptions on $\varphi$ and $\gamma$.