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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair

IMPACT FACTOR 2018: 1.650

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Volume 5, Issue 4


Averaging of Attractors and Inertial Manifolds for Parabolic PDE With Random Coefficients

Igor D. Chueshov / Björn Schmalfuß
Published Online: 2016-03-10 | DOI: https://doi.org/10.1515/ans-2005-0402


The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

Keywords: averaging principle; attractors; inertial manifolds; random dynamical systems

About the article

Received: 2005-07-04

Published Online: 2016-03-10

Published in Print: 2005-11-01

Citation Information: Advanced Nonlinear Studies, Volume 5, Issue 4, Pages 461–492, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2005-0402.

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© 2016 by Advanced Nonlinear Studies, Inc..

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