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Asymptotics for periodic systems

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Details

Original languageEnglish
Pages (from-to)7152-7172
JournalJournal of Differential Equations
Volume266
Issue number11
Early online date2018
DOIs
Publication statusPublished - May 2019
Publication typeA1 Journal article-refereed

Abstract

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the one-dimensional wave equation with periodic damping.

ASJC Scopus subject areas

Keywords

  • Damped wave equation, Evolution family, Non-autonomous system, Periodic, Rates of convergence, Ritt operator

Publication forum classification

Field of science, Statistics Finland