RANDOM ATTRACTORS FOR NON-AUTONOMOUS STOCHASTIC WAVE EQUATIONS WITH STRONG DAMPING AND ADDITIVE NOISE ON <inline-formula><tex-math id="M1">$ {\mathbb{R}}^{N} $</tex-math></inline-formula>

Abstract

This paper investigates the long-time behavior of a stochastic strongly damped wave equation with additive noise on $\mathbb{R}$^<i>N</i>. We establish that there exists a unique pullback random attractor for the equation in natural space $\mathcal{H}=H^1(\mathbb{R}^{N})\times L^2(\mathbb{R}^{N})$ with the nonlinearity <i>g</i>(<i>x</i>,<i>u</i>) being of optimal subcritical growth <i>p</i>: 1≤<i>p</i><<i>p</i>^* ≡ <i>N</i>+2/(<i>N</i>-2)(<i>N</i> ≥ 3). In addition, we get the upper semicontinuity of the pullback random attractor as the intensity of noise goes to zero.