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Invariant manifolds for stochastic wave equations. (English) Zbl 1113.37056
Summary: We consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.

37L55 Infinite-dimensional random dynamical systems; stochastic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
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