The Existence, Regularity And Measurability Of Pullback Attractors For Multi-valued Non-autonomous Non-compact Random Dynamical System

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of proving asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to establish the existence of pullback attractors for the multi-valued non-compact random dynamical systems. We first present a sufficient and necessary condition for existence of pullback attractors of multi-valued non-compact random dynamical systems. For random sys-tems containing periodic or almost periodic deterministic forcing terms, we show the pullback attractors are also periodic or almost periodic under certain conditions.Secondly, we study pullback attractors of reaction-diffusion equations on an un-bounded domain with non-autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold. We then prove the existence of pullback attractors in L2(Rn), Lp(Rn) and H1(Rn) for multi-valued non-compact ran-dom dynamical systems associated with the reaction-diffusion equations on Rn. More-over, we show that the DH-pullback attractor, (DH,DLp)-pullback attractor and the (DH, Dv)-pullback attractor are identical in the set inclusion-relation sense. We ob-tain some properties concerning the map ωâ†'Φ(l,Ï„, ω, K(Ï„, ω)), where K is the closed measurable pullback absorbing set, and then show the measurability of the pullback attractor.Then, we prove the existence and the measurability of pullback attractors for the multi-valued non-compact random dynamical systems associated with the semi-linear degenerate parabolic unbounded delay equations with both deterministic and random external terms.Finally, the existence and the measurability of pullback attractors for the multi-valued non-compact random dynamical systems generated by stochastic damped non-autonomous delay wave equations with linear memory and additive white noise are also addresses when the nonlinearity has a subcritically growing exponent.