Random attractors for stochastic time-dependent damped wave equation with critical exponents

We study the asymptotic behavior of solutions of a stochastic time-dependent damped wave equation. With the critical growth restrictions on the nonlinearity $ f $ and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in $ H^1_0(U)\times L^2(U) $. In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.