Stability of constant steady states of a chemotaxis model

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.

20 pages