Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

TL;DR

This paper investigates the asymptotic convergence rate of solutions to the two-dimensional Keller-Segel model with small mass, establishing exponential convergence to a stationary state under certain conditions.

Contribution

It provides a new proof of exponential convergence rates for solutions with small mass in the Keller-Segel system using self-similar variables.

Findings

Solutions converge exponentially to stationary state

Convergence rate depends on the mass being below a threshold

Provides insights into intermediate asymptotics of the model

Abstract

The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8\pi $, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.