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Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients. (English) Zbl 1092.37051
Summary: The averaging method has been used to study random or nonautonomous 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.

MSC:
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
37L55 Infinite-dimensional random dynamical systems; stochastic equations
35B42 Inertial manifolds
37H10 Generation, random and stochastic difference and differential equations
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35B41 Attractors
35K55 Nonlinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
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