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Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise

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This work is supported by National Natural Science Foundation of China grant 11571283

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  • 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.

    Mathematics Subject Classification: Primary: 35B41, 60H15; Secondary: 35R60.

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