Dynamics for a stochastic reaction–diffusion equation with additive noise

In this paper, we present a new scheme to study the dynamics of a stochastic reaction–diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p⩾2p⩾2. Firstly we use this scheme to establish some new estimates, higher-order integrability of the difference of the solutions near the initial time, instead of using the usual estimates about higher regularities and higher-order integrability of solutions. Secondly we verify that the attraction in the usual (L2(Q),L2(Q))(L2(Q),L2(Q))DD-pullback random attractor indeed can be L2+δL2+δ-norm for any δ∈[0,∞)δ∈[0,∞); and the solutions of the equation are continuous in H01(Q) with respect to initial data. Thirdly we obtain the existence of pullback random attractor in H01(Q) as an application of the continuity. Even for the corresponding deterministic cases, the results and methods introduced in this paper are new.