Stability of constant steady states of a chemotaxis model

Abstract The Cauchy problem for the parabolic–elliptic Keller–Segel system in whole n -dimensional space is studied. For this model, every constant $$A \in {\mathbb {R}}$$ A∈R a stationary solution. main goal of work to show that < 1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">A<1 stable steady state while > xmlns:mml="http://www.w3.org/1998/Math/MathML">A>1 unstable. Uniformly local Lebesgue spaces are used order deal with solutions do not decay at spatial variable on unbounded domain.