H1H1-random attractors for stochastic reaction–diffusion equations with additive noise

In this paper, a new theorem obtained by Li and Guo [Y. Li, B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction–diffusion equations, J. Differential Equations 245(7) (2008) 1775–1800] is employed to some reaction–diffusion equations with additive noise. We prove the unique existence of DD-random attractors in H01 for the corresponding random dynamical system. We also obtain the existence of a unique random equilibrium for the generated random dynamical system under some restrictive assumption on the coefficients.