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Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE

This work is supported by National Natural Science Foundation of China grant 11571283 and Natural Science Foundation of Guizhou Province (KY[2016]103).
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  • This paper is concerned with the robustness of a pullback attractor as the time tends to infinity. A pullback attractor is called forward (resp. backward) compact if the union over the future (resp. the past) is pre-compact. We prove that the forward (resp. backward) compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive (resp. negative) infinity, and also obtain the minimal limit-set. We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity. Some criteria for such robustness are established when the evolution process is forward or backward omega-limit compact. Those theoretical criteria are applied to prove semi-uniform compactness and robustness at infinity in pullback dynamics for a Ginzburg-Landau equation with variable coefficients and a forward or backward tempered nonlinearity.

    Mathematics Subject Classification: Primary: 37L55, 37L30; Secondary: 35B41.

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