H1-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises

In this paper, we consider the stochastic reaction-diffusion equation with the general growth and dissipative assumptions of the forcing terms. Based on some new estimates, the flattening condition in H-0(1) for the corresponding random dynamical system is proved and then the random dynamics for this is captured by a random attractor in H-0(1). The existence of a unique random equilibrium is obtained for some negative coefficients. (C) 2013 Elsevier B.V. All rights reserved.