Upper semi-continuity and regularity of random attractors on pp-times integrable spaces and applications

A sufficient condition for a family of random attractors to be upper semi-continuous and regular is obtained when the state space is a pp-times integrable space with p>2p>2. It is shown that both upper semi-continuity and regularity under the topology of pp-norms do not rely on continuity, convergence and absorption of the associated random dynamical systems in LpLp since the corresponding L2L2-properties can offer all. As an application of the abstract result, it is shown that the family of random attractors for the stochastic reaction–diffusion equations on a unbounded domain is upper semi-continuous and regular at any point under the topology of pp-norms.