Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles. (English) Zbl 1360.37135
Summary: 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 \((L^2, H_0^1)\)-pullback attractor for a stochastic complex Ginzburg-Landau equation. A spectrum decomposition method is employed to overcome the lack of Sobolev compactness embeddings in \(H_0^1\).
MSC:
| 37H05 | General theory of random and stochastic dynamical systems |
| 35Q56 | Ginzburg-Landau equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
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