Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients.

*(English)*Zbl 1092.37051Summary: 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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\textit{I. D. Chueshov} and \textit{B. Schmalfuß}, Adv. Nonlinear Stud. 5, No. 4, 461--492 (2005; Zbl 1092.37051)

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