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 ∈ 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.