Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$

Abstract

The existence of a random attractor in H 1 (ℝ 3 ) x L 2 (ℝ 3 ) is proved for the damped semilinear stochastic wave equation defined on the entire space ℝ 3 . The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.