On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics
type: Item
instance of: scholarly article
title: On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics
zbMATH Open document ID: 1419.35096
author: Xie Li
author: Yilong Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-12-20T00:00:00Z
review text: The authors study the two-species chemotaxis system with Lotka-Volterra competition \[ \begin{cases} \partial_t u = \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1-u- a_1 v), & \quad (x,t) \in \Omega \times (0,\infty),\\ \partial_t v = \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v (1-v- a_2 u), & \quad (x,t) \in \Omega \times (0,\infty), \\ \partial_t w = \Delta w - \lambda w + b_1 u + b_2 v, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and sufficiently regular inital data which are nonnegative and nonzero. Here $\Omega \subset \mathbb{R}^3$ is a smooth, bounded domain and $\chi_i, \mu_i, a_i, b_i, \lambda$ are positive constants. $u$ and $v$ denote the densities of two populations which compete for resources and move toward increasing concentrations of the chemoattractant $w$ which they produce themselves. \par The authors prove the existence of a global and bounded classical solution if $\frac{\mu_i}{b_i}$ is suffiently large as compared to $\chi_1$ and $\chi_2$ for $i=1,2$. \par An important ingredient of the proof is the derivation of a differential inequality which can be seen as an entropy type functional and allows to prove the boundedness of appropriate $L^p$-norms of the solution components.
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A09
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 6996196
zbMATH Keywords: chemotaxis
zbMATH Keywords: global existence
zbMATH Keywords: boundedness
zbMATH Keywords: Lotka-Volterra competitive kinetics
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.10.093
OpenAlex ID: W2898969356
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Wikidata QID: Q129013593
DOI: 10.1016/J.JMAA.2018.10.093
description: scientific article; zbMATH DE number 6996196