Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller--Segel Model with Signal-dependent Motilities for Local Sensing Chemotaxis

TL;DR

This paper analyzes a degenerate Keller--Segel chemotaxis model with signal-dependent motility, establishing boundedness and exponential stabilization of solutions in higher dimensions, which advances understanding of such systems.

Contribution

It introduces new techniques to prove boundedness and exponential stabilization for a Keller--Segel model with degenerate, signal-dependent motility functions, improving previous results.

Findings

Established a time-independent upper bound for cell concentration.

Proved uniform-in-time boundedness of solutions.

Demonstrated exponential convergence to steady states.

Abstract

In this paper we consider the initial Neumann boundary value problem for a degenerate Keller--Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos--Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing of an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in \cite{Anh19,FJ20a} and the exponential stabilization is obtained for the first time.