On a Repulsion Keller--Segel System with a Logarithmic Sensitivity
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 . 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.
Cite
@article{arxiv.2007.02546,
title = {On a Repulsion Keller--Segel System with a Logarithmic Sensitivity},
author = {Jie Jiang},
journal= {arXiv preprint arXiv:2007.02546},
year = {2020}
}