Random dynamics of dispersive-dissipative wave equations driven by nonlinear colored noise

This paper is devoted to the asymptotic behavior of solutions to a class of non-autonomous random dispersive-dissipative wave equations driven by nonlinear colored noise defined on unbounded domains. We first prove the existence of pullback random attractors of the random wave equation driven by nonlinear colored noise, and then establish the upper semi-continuity of the random attractors to a class of random wave equations driven by linear operator-type colored noise as the correlation time tends to zero. The operator-type noise is unbounded which is multiplied by a Laplace operator. Both methods of spectral decomposition and uniform tail-estimates are combined to achieve the pullback asymptotic compactness of the solutions in order to overcome the difficulty arising from the lack of compact Sobolev embeddings on unbounded domains.