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Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. (English) Zbl 1411.35048

Summary: A non-autonomous random attractor is called backward compact if its backward union is pre-compact. We show that such a backward compact random attractor exists if a non-autonomous random dynamical system is bounded dissipative and backward asymptotically compact. We also obtain both backward compact and periodic random attractor from a periodic and locally asymptotically compact system. The abstract results are applied to the sine-Gordon equation with multiplicative noise and a time-dependent force. If we assume that the density of noise is small and that the force is backward tempered and backward complement-small, then, we obtain a backward compact random attractor on the universe consisted of all backward tempered sets. Also, we obtain both backward compactness and periodicity of the attractor under the assumption of a periodic force.

MSC:

35B41 Attractors
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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