Global Boundedness Induced by Asymptotically Non‐Degenerate Motility in a Fully Parabolic Chemotaxis Model with Local Sensing

ABSTRACT A fully parabolic chemotaxis model of Keller–Segel type with local sensing is considered. The system features a signal‐dependent asymptotically non‐degenerate motility function, which accounts for a repulsion‐dominated chemotaxis. Global boundedness of classical solutions is proved for an initial Neumann boundary value problem of the system in any space dimension. In addition, stabilization toward the homogeneous steady state is shown, provided that the motility is monotone non‐decreasing. The key steps of the proof consist of the introduction of several auxiliary functions and a refined comparison argument, along with uniform‐in‐time estimates for a functional involving nonlinear coupling between the unknowns and auxiliary functions.