Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on \(\mathbb{R}^N\)
zbMATH Open document ID: 1501.35073
author: Wenxian Shen
author: Shuwen Xue
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2022-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2107.01551
review text: The authors consider the parabolic-parabolic chemotaxis model with logistic source \[ \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla b) + u(a - bu), \\ \tau v_t = \Delta v - \lambda v + \mu v \end{cases} \] in \(\mathbb R^N\) with \(\tau = 1\) and positive parameters \(\chi, a, b, \lambda, \mu\), and prove that that the spreading speed for solutions to nontrivial initial data with finite support is \(2\sqrt a\), provided \(\chi < \frac{4b}{N\mu}\). Since the spreading speed for the Fisher-KPP equation, i.e.\ for the first equation in the system above with \(\chi = 0\), is also \(2\sqrt a\), this shows that (at least when \(\chi\) is small or when \(b\) is large) chemotaxis has no effect on the spreading speed. Thereby, the authors transfer results such as [\textit{R. B. Salako} et al., J. Math. Biol. 79, No. 4, 1455--1490 (2019; Zbl 1479.35116)] for the parabolic-elliptic version, i.e\ the system above with \(\tau = 0\), to the fully parabolic setting. Upper bounds on the spreading speed are obtained by applying the comparison principle to \(w = u + \frac{\chi}{2\mu} |\nabla v|^2\) which fulfills \(w_t \le \Delta w + a w\). This technique makes essential use of the assumption \(\tau=1\).
reviewed by: Mario Fuest
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 7593144
zbMATH Keywords: chemotaxis
zbMATH Keywords: spreading speeds
zbMATH Keywords: logistic source
MaRDI profile type: Publication
OpenAlex ID: W3181345309
arXiv ID: 2107.01551
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DOI: 10.3934/DCDSS.2022074
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