Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients
type: Item
instance of: scholarly article
title: Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients
zbMATH Open document ID: 0733.35008
author: Qing Fang
published in: Hiroshima Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The author deals with a reaction-diffusion system in the dumbbell shaped domain. From the condition that the coefficients are very large, he concludes that the solution approaches a locally constant function except for the very thin region of the dumbbell domain. From the dynamical system point of view, this means that some specific finite dimensional set attracts all solutions eventually. This set is sometimes called an inertial manifold. Such a research has been done first by \textit{Y. Morita} [J. Dyn. Differ. Equations 2, No.1, 69-115 (1990; Zbl 0702.35129)]. The present author obtains the reduced equation on the inertial manifold more precisely. To do this, he obtains a very elaborate characterization of the second eigenvalue of the dumbbell shaped domain (Neumann B. C.). He also gives an application to the competition system and obtains a result corresponding to the Hale and Vegas's famous work on a single equation [see \textit{J. K. Hale} and \textit{J. Vegas}, Arch. Ration. Mech. Anal. 86, 99-123 (1984; Zbl 0569.35048)].
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 4212557
zbMATH Keywords: attractor
zbMATH Keywords: domain-dependency
zbMATH Keywords: dumbbell shaped domain
zbMATH Keywords: locally constant function
zbMATH Keywords: inertial manifold
reviewed by: Shuichi Jimbo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206129049
OpenAlex ID: W1558653997
Wikidata QID: Q128900080
DOI: 10.32917/HMJ/1206129049
description: scientific article; zbMATH DE number 4212557