H1H1-random attractors and random equilibria for stochastic reaction–diffusion equations with multiplicative noises

• The stochastic reaction–diffusion equations with general forcing function as well as multiplicative are considered.
• The existence of a pullback random attractors is established for the generated RDS.
• The unique random equilibria exist for a wide range of coefficients.

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 H01 for the corresponding random dynamical system is proved and then the random dynamics for this is captured by a random attractor in H01. The existence of a unique random equilibrium is obtained for some negative coefficients.