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 $$A \in {\mathbb {R}}$$ A ∈ R is a stationary solution. The main goal of this work is to show that $$A < 1$$ A < 1 is a stable steady state while $$A > 1$$ 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.

Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

“…Analogous result is also obtained by Cygan-Karch-Krawczyk-Wakui [7], where they consider the stability of a constant solution. A natural question for the Keller-Segel system under such a setting is whether the singular limit problem (1.2) can be justified in such a locally uniform class of solutions.…”

“…Proof of Proposition 2. 7 In order to show (2.6), it suffices to consider the case of s 1 = s and s 0 = 0 for s > 0. By (2.4), we have…”