Stability of constant steady states of a chemotaxis model

TL;DR

This paper analyzes the stability of constant steady states in a chemotaxis model, showing that states below a certain threshold are stable while those above are unstable, using advanced functional spaces for unbounded domains.

Contribution

It establishes the stability criteria for constant steady states in the Keller--Segel chemotaxis system on unbounded domains.

Findings

States with A < 1 are stable.

States with A > 1 are unstable.

Uses uniformly local Lebesgue spaces for analysis.

Abstract

The Cauchy problem for the parabolic--elliptic Keller--Segel system in the whole $n$-dimensional space is studied. For this model, every constant $A \in \mathbb{R}$ is a stationary solution. The main goal of this work is to show that $A < 1$ is a stable steady state while $A > 1$ is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.