Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis
zbMATH Open document ID: 1400.35162
author: Yumi Yahagi
published in: Tokyo Journal of Mathematics
publication date: 2018-10-26T00:00:00Z
review text: Let \((u,v)\) be a global classical solution to the one-dimensional Keller-Segel chemotaxis system \[ \begin{aligned} \partial_t u & = \partial_x \left( \partial_ x u - \chi u \partial_x v \right) \;\text{ in }\; (0,\infty)\times I\;, \cr \partial_t v & = \partial_x^2 v - \gamma v + \alpha u \;\text{ in }\; (0,\infty)\times I\;, \\ \partial_x u & = \partial_x v = 0 \;\text{ on }\; (0,\infty)\times \partial I\;, \\ (u,v)(0) & = (\bar{u}, \bar{v}) \;\text{ in }\; I\;, \end{aligned} \] where \(I=(a,b)\) is a bounded interval of \(\mathbb{R}\), \(\chi\), \(\gamma\), and \(\alpha\) are positive real numbers, and \((\bar{u}, \bar{v})\) are positive initial conditions in \(W^{1,2}(I)\) with \(M(b-a)=\|\bar{u}\|_{L^1}\). The author proves the following conditional theorem: if there is \(t_*>0\) such that \[ \chi C \|\partial_x v(t)\|_{L^\infty} < 1\;, \qquad t\geq t_*\;, \] the constant \(C\) being that in Poincaré's inequality, and if \[ \lim_{t\to\infty} \|\partial_x^2 u(t)\|_{L^2} = 0\;, \] then \((u(t),v(t))\) converges to \((M,\alpha M/\gamma)\) in \(L^2(I;\mathbb{R}^2)\) as \(t\to\infty\). No examples of initial data to which the previous result may apply are given. Moreover, the intermediate results are either well-known and standard properties of linear parabolic equations or classical differential inequalities and their proofs could have been omitted.
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 6966863
zbMATH Keywords: Keller-Segel system
zbMATH Keywords: large time behaviour
reviewed by: Philippe Laurençot
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1502179267
OpenAlex ID: W2785019731
cites work: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues
cites work: Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces
cites work: Q3531367
cites work: Q4399897
cites work: Elliptic partial differential equations of second order
cites work: A user's guide to PDE models for chemotaxis
cites work: Initiation of slime mold aggregation viewed as an instability
cites work: Q3107912
cites work: Q4824324
cites work: Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model
cites work: Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening
DOI: 10.3836/TJM/1502179267
description: scientific article; zbMATH DE number 6966863