Non--autonomous and random attractors for delay random semilinear equations without uniqueness

Abstract

Abstract. We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by R − . In particular, we do not assume the uniqueness of solutions of these equations.1. Introduction. The intention of this article is to study the asymptotic behaviour of multi-valued non-autonomous and random dynamical systems. The long-time behaviour of these systems can be expressed by terms like pullback attractor and random attractor. The theories of these attractors are now well established as have been extensively developed over the last one and a half decades (see, e.g. Caraballo et al.

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

“…The following lemma is concerning the properties for a random set, which will be needed in the following arguments. See [8,12,22].…”

“…In general, the continuity and measurability of a solution of an SPDE in the initial space are easy to check, and therefore to derive a (measurable) random attractor in such a space we only need to look for the compact condition and the absorption property, which has been extensively studied ever since [14,15,37] first introduced the concept of random attractor. For example, we may refer to [8,9,10,19] for the autonomous case and [6,43,44,45,48] for the non-autonomous case. However, it is challenging to prove the continuity and measurability of solution of an SPDE in the corresponding regular spaces, see the statements in the introductions of the literature [7,8,31].…”

“…For example, we may refer to [8,9,10,19] for the autonomous case and [6,43,44,45,48] for the non-autonomous case. However, it is challenging to prove the continuity and measurability of solution of an SPDE in the corresponding regular spaces, see the statements in the introductions of the literature [7,8,31]. Therefore, the measurability of a random attractor in the regular space is not explicitly studied in the literature, although the existence of bi-spatial random attractor has been obtained for a large volume of stochastic partial equations (cf.…”

“…It is well known that the existence theorem of pullback attractor for the deterministic dynamical system can be applied to analyse the long-time dynamics of random dynamical system with some modifications. However, in the case of random (stochastic) differential equations, we need additional measurability properties to assure that the solutions are well-posed [8]. This leads to an additional and great difference in comparison with the non-random case.…”

“…In particular, the properties of the Ornstein-Uhlenbeck process over Ω k are substantially employed. The reader is referred to [8,9,10,47] for the idea to restrict Ω to Ω k .…”