Upper semi-continuity and regularity of random attractors on p-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 p-times integrable space with p>2. It is shown that both upper semi-continuity and regularity under the topology of p-norms do not rely on continuity, convergence and absorption of the associated random dynamical systems in Lp since the corresponding L2-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 p-norms.