Asymptotics for periodic systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Paunonen, Lassi¹  Seifert, David² 

[1] Tampere Univ Technol, Dept Math, POB 553, FIN-33101 Tampere, Finland

[2] St Johns Coll, Oxford OX1 3JP, England

关键词: Rates of convergence;    Non-autonomous system;    Periodic;    Evolution family;    Ritt operator;    Damped wave equation;   

DOI  :  10.1016/j.jde.2018.11.028

来源: Elsevier

【 摘 要 】

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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free