Global boundedness induced by asymptotically non-degenerate motility in a fully parabolic chemotaxis model with local sensing

TL;DR

This paper proves global boundedness and stabilization of solutions in a chemotaxis model with local sensing and signal-dependent motility, highlighting conditions for non-degenerate motility functions.

Contribution

It introduces a novel approach to establish global boundedness in a fully parabolic chemotaxis system with local sensing and non-degenerate motility.

Findings

Global boundedness of solutions proved in any space dimension

Stabilization towards homogeneous steady state under monotone non-decreasing motility

Refined comparison and auxiliary functions used in the proof

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 towards 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.