Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.