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.