Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Abstract

We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al. (Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zeromemory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory.

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

“…The measurability of R λ (•) implies that K λ (•) is a random set in X. By (20) in Lemma 5.1, for D ∈ D and ω ∈ Ω, there is…”

“…The continuity of random attractors includes both upper and lower semi-continuities. The upper semi-continuity describes the nonexplosive phenomenon, which is expected to hold widely, see the abstract results in [5,29,31] and many applications in [7,16,17,18,20,35,38,42,43]. The lower semi-continuity describes the non-implosive phenomenon, which is extremely hard to prove in partial differential equations (PDE) as pointed out by Carvalho et al [6, page 65].…”