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.
Highlights
There are several mathematical works on the chemotaxis model introduced by Keller and Segel [15]
For each constant A ∈ R, the couple (u, ψ) = (A, A) is a stationary solution of system (1.1), and since the domain is unbounded, it does not belong to any Lebesgue L p-space with p ∈ [1, ∞)
We consider a constant stationary solution (u, ψ) = (A, A) with A ∈ [0, 1), and we show in Theorem 2.3 that a small L p-perturbation of such an initial datum gives a global-in-time solution which converges toward (A, A) as t → ∞
Summary
There are several mathematical works on the chemotaxis model introduced by Keller and Segel [15]. Where u = u(t, x) denotes the density of cells and ψ = ψ(t, x) is a concentration of chemoattractant In these equations, all constant parameters are equal to one for simplicity of the exposition. System (1.1) is already studied in the whole space e.g. in the papers [4,5,6,13,14,16,17,19], where several results either on a blow-up or on a large-time behavior of solutions have been obtained. The paper [10] describes dynamics near an unstable constant solution to the classical parabolic–parabolic Keller–Segel model in a bounded domain, and obtained results are interpreted as an early pattern formation.
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