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Dynamics of non-autonomous stochastic dissipative Hamiltonian amplitude wave equations with rapidly oscillating force. (English) Zbl 1407.35032

Summary: This paper considers the limiting behavior of pullback attractors for the non-autonomous stochastic dissipative Hamiltonian amplitude wave equation with rapidly oscillating force \(g^\epsilon(x,t)\) defined on a real line. The presence of the complex valued term \(iu_x\) brings some difficulties even to prove the induced system is pullback absorbing. Thus, by choosing a new number \(\delta >0\), some new uniform asymptotic estimates are used to prove the existence of pullback attractors in \(H^1(\mathbb R)\times L^2(\mathbb R)\), as well as to show the upper semicontinuity of the family of oscillating attractors when \(g^\epsilon (x,t)\) tends to the average \(g^0(x,t)\) as the density of noise \(\epsilon\) approaches to zero.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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