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Uniform Global Attractors for Nonautonomous Evolution Inclusions

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Advances in Dynamical Systems and Control

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 69))

Abstract

In this note, we prove the existence and provide basic structure properties of compact (in the natural phase space) uniform global attractor for all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions. The obtained results allow to reduce the problem of the complete qualitative investigation of various nonlinear systems into the “small” (compact) part of the natural phase space.

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Notes

  1. 1.

    That is, \(V_i\) is a real reflexive separable Banach space continuously and densely embedded into a real Hilbert space H, H is identified with its topologically conjugated space \(H^*\), \(V_i^*\) is a dual space to \(V_i\). So, there is a chain of continuous and dense embeddings: \(V_i\subset H\equiv H^*\subset V_i^*\) (see, e.g., Gajewski, Gröger, and Zacharias [1, Chap. I]).

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Acknowledgments

This work was partially supported by the Ukrainian State Fund for Fundamental Researches under grant GP/F66/14921, and by the National Academy of Sciences of Ukraine under grant 2284.

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Authors and Affiliations

  1. National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy Ave. 37, Build. 1, Kyiv, 03056, Ukraine

    Mikhail Z. Zgurovsky

  2. Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy Ave. 37, Build. 35, Kyiv, 03056, Ukraine

    Pavlo O. Kasyanov

Authors
  1. Mikhail Z. Zgurovsky
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  2. Pavlo O. Kasyanov
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Correspondence to Pavlo O. Kasyanov .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. Kyiv Polytechnic Institute, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

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Zgurovsky, M.Z., Kasyanov, P.O. (2016). Uniform Global Attractors for Nonautonomous Evolution Inclusions. In: Sadovnichiy, V., Zgurovsky, M. (eds) Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-40673-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-40673-2_3

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