Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction–diffusion equations on an unbounded domain

In this paper we study pullback attractors of reaction–diffusion equations on an unbounded domain with non-autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold. We first present the existence and structure of pullback attractors of multi-valued non-compact random dynamical systems. Then we prove the existence of pullback attractors in L2(Rn), Lp(Rn) and H1(Rn) for multi-valued non-compact random dynamical systems associated with the reaction–diffusion equations on Rn, and the identical relation of pullback attractors in different spaces is also provided. In particular, the measurability of pullback attractors is established for reaction–diffusion equations on Rn.