Nonlinear stability for dissipative nonlinear evolution equations with ellipticity
type: Item
instance of: scholarly article
title: Nonlinear stability for dissipative nonlinear evolution equations with ellipticity
zbMATH Open document ID: 0932.35022
author: Huijiang Zhao
author: Shaoqiang Tang
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-28T00:00:00Z
review text: The paper deals with the coupled system \[ \psi_t=-(1-\alpha)\psi-\sigma \theta_x +\alpha\psi_{xx}, \] \[ \theta_t=-(1-\alpha)\theta+\nu \psi_x +2\psi \theta_{x}+\alpha\theta_{xx}, \] where \(\alpha\) and \(\nu\) are positive numbers. This set of equations singles out the essential mechanism of nonlinear interaction between ellipticity and dissipation. It is proved that the considered system is linearly stable if and only if \[ \nu< 4\alpha (1- \alpha). \] Moreover, if the initial data is small enough, the optimal decay rate has been obtained. Theorem 5 of the paper is in some sense closely related to Theorem 15.5.1 from the book of the reviewer [Stability of finite and infinite dimensional systems, Kluwer Academic Publishers, Boston-Dordrecht-London (1998; Zbl 0916.93002)].
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1289966
zbMATH Keywords: optimal decay rate
zbMATH Keywords: nonlinear interaction between ellipticity and dissipation
reviewed by: Michael I. Gil'
MaRDI profile type: Publication
OpenAlex ID: W2080871647
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DOI: 10.1006/JMAA.1999.6316
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description: scientific article; zbMATH DE number 1289966