Dynamics for the Stochastic Reaction-di usion Equation with Distribution Derivatives and Multiplicative Noise on ℝn

Recently the existence of random attractors for a stochastic reaction-diffusion equation without distribution derivatives has been studied by many authors, the existence of global attractors was established for the deterministic reaction-diffusion equation with distribution derivatives has been studied by Sun et al. in [1, 2]. In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives and multiplicative noise defined on unbounded domains. In order to obtain the asymptotic compactness of the random dynamical system, we make use of a priori estimates for far-field values of solutions as well as the cut-off technique.