Attractors of resonance wave type equations: Discontinuous oscillations
type: Item
instance of: scholarly article
title: Attractors of resonance wave type equations: Discontinuous oscillations
zbMATH Open document ID: 0841.35071
DOI: 10.1007/BF02110556
published in: Mathematical Notes
publication date: 1996-02-01T00:00:00Z
review text: In a previous paper [Russ. Acad. Sci., Sb. Math. 78, No.2, 367-378 (1994; Zbl 0817.35038); translation from Mat. Sb. 184, No. 3, 121-136 (1993)], the author defined the concepts of orbital exponential stability and of quasinormal form of a boundary problem. In this paper, he investigates the boundary value problem: \[ {\partial^2 u\over \partial t^2}- \varepsilon {\partial\over \partial t} \Biggl(u+ a^2 {\partial^2 u\over \partial x^2}\Biggr)+ u^2 {\partial u\over \partial t}= {\partial^2 u\over \partial x^2};\;u|_{x= 0}= u|_{x= 1}= 0\tag{1} \] with \(0< a\pi< 1\). He states that, for \(\varepsilon\) small enough, problem (1) has an orbitally exponential stable cycle \(U\) whose asymptotic representation is of the form \[ U= \varepsilon^{1/2}[v(t- x)- v(t+ x)]+ O(\varepsilon^{3/2}), \] where the 2-periodic function \(v\) is the equilibrium state of the quasinormal form of problem (1). The result is proved by modifying the constructions of loc. cit.
reviewed by: Denise Huet
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 830687
zbMATH Keywords: attractors
zbMATH Keywords: resonance wave type equations
zbMATH Keywords: discontinuous oscillations
zbMATH Keywords: quasinormalization method
zbMATH Keywords: orbital exponential stability
author: Ju. S. Kolesov
MaRDI profile type: Publication
cites work: Q3049314
full work available at URL: https://doi.org/10.1007/bf02110556
OpenAlex ID: W1998284627
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description: scientific article; zbMATH DE number 830687