Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles

In this work, we establish several criteria for the existence as well as the upper semi-continuity of bi-spatial attractors under a closedness condition, which dramatically weakens the usual requirement on the continuity of the cocycle. It is also shown that, though the continuity plays a less important role in the study of attractors, it is impossible to establish an existence criteria for common attractors for systems without any continuity-like properties. However, for such “bad” systems, one can expect a mini attractor, which is shown adequate well to depict the asymptotic behavior of non-continuous systems. Finally, we study the (L2,H01)-pullback attractor for a stochastic complex Ginzburg–Landau equation. A spectrum decomposition method is employed to overcome the lack of Sobolev compactness embeddings in H01.