Stability of constant steady states of a chemotaxis model

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.