No pattern formation in a quasilinear chemotaxis model with local sensing

Convergence to spatially homogeneous steady states is shown for a chemotaxis model with local sensing and possibly nonlinear diffusion when the intrinsic diffusion rate φ dominates the inverse of the chemotactic motility function γ, in the sense that (φγ)'≥ 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 φγ is non-decreasing. The convergence proof relies on a detailed study of the dissipation of the Liapunov functional and requires additional technical assumptions on φ and γ.