(L2,Lp)(L2,Lp)-random attractors for stochastic reaction–diffusion equation on unbounded domains

In this paper, the existence of (L2,Lp)(L2,Lp)-random attractor is established for a stochastic reaction–diffusion equation on the whole space RNRN. This random attractor is a compact and invariant tempered set which attracts every tempered random subset of L2L2 in the topology of LpLp. The nonlinearity ff is supposed to satisfy some growth of arbitrary order p−1p−1, where p≥2p≥2. The (L2,Lp)(L2,Lp)-asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions.