On a Repulsion Keller--Segel System with a Logarithmic Sensitivity

TL;DR

This paper analyzes a Keller--Segel system with logarithmic sensitivity, proving existence, regularity, and exponential convergence of solutions in different dimensions, using a novel approach based on stability and dissipative properties.

Contribution

It introduces a new method to establish eventual regularity and exponential convergence for solutions of the Keller--Segel system with logarithmic sensitivity.

Findings

Existence of classical solutions in 2D

Existence of weak solutions in 3D

Weak solutions become regular after finite time

Abstract

In this paper, we study the initial-boundary value problem of a repulsion Keller--Segel system with a logarithmic sensitivity modeling the reinforced random walk. By establishing an energy-dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoys an eventual regularity property, i.e., it becomes regular after certain time $T>0$. An exponential convergence rate toward the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author \cite{J19} to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.