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 >= 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 (L-2 (Q), L-2 (Q)) D-pullback random attractor indeed can be L2+delta-norm for any delta is an element of [0, infinity); and the solutions of the equation are continuous in H-0(1) (Q) with respect to initial data. Thirdly we obtain the existence of pullback random attractor in H-0(1) (Q) as an application of the continuity. Even for the corresponding deterministic cases, the results and methods introduced in this paper are new. (C) 2015 Elsevier Inc. All rights reserved.