Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay.
type: Item
instance of: scholarly article
title: Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay.
zbMATH Open document ID: 1080.35168
author: A. V. Rezunenko
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116151
review text: By constructing inertial manifolds with delay (IMD) for dissipative systems of second order in time, the author studies the different asymptotic properties of solutions. The author gives the proof for the retarded case and notice that even for the nonretarded case it has not been done before in the context of systems of second order in time. Using IMD, the author constructs approximate inertial manifolds; containing all the stationary solutions. He improves the result of [\textit{I. D. Chueshov} and \textit{A. V. Rezounenko}, C. R. Acad. Sci., Paris, Sér. I 321, 607--612 (1995; Zbl 0845.35129)] on the finite number of essential modes and gives a new characterization of the \(\mathcal{K}\)-invariant manifold constructed in [\textit{M. Taboada} and \textit{Y. You}, J. Differ. Equations 114, No. 2, 337--369 (1994; Zbl 0815.34067)]. In contrast to the case of parabolic equations with delay [\textit{L. Boutet de Monvel, I. D. Chueshov} and \textit{A. V. Rezounenko}, Nonlinear Anal., Theory Methods Appl. 34, No. 6, 907--925 (1998; Zbl 0954.34064); \textit{A. V. Rezounenko}, Discrete Contin. Dyn. Syst. 6, No. 4, 829--840 (2000; Zbl 1011.37046); J. Math. Anal. Appl. 282, No. 2, 614--628 (2003; Zbl 1039.35133)], the investigations of this paper are crucially based upon a special choice of the norm introduced in [\textit{X. Mora}, Pitman Res. Notes Math. Ser. 155, 172--183 (1987; Zbl 0642.35061)].
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35B42
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 37L65
zbMATH DE Number: 2162433
zbMATH Keywords: approximate inertial manifolds
zbMATH Keywords: inertial manifolds with delay
zbMATH Keywords: retarded nonlinear partial differential equations
MaRDI profile type: Publication
OpenAlex ID: W2315544731
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DOI: 10.5802/AIF.2058
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162433