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Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System

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Continuous and Distributed Systems

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 211))

Abstract

In this paper we study structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients we prove that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in \(L^{\infty }(\varOmega )\cap H_0^1(\varOmega )\).

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Acknowledgments

The first two authors were partially supported by the Ukrainian State Fund for Fundamental Researches under grants GP/F44/076 and GP/F49/070.

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka St, Kyiv, 01601, Ukraine

    Oleksiy V. Kapustyan

  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 & Mikhail Z. Zgurovsky

  3. Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n, 03202 , Elche, Spain

    José Valero

Authors
  1. Oleksiy V. Kapustyan
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  2. Pavlo O. Kasyanov
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  3. José Valero
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  4. Mikhail Z. Zgurovsky
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Corresponding author

Correspondence to Oleksiy V. Kapustyan .

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Editors and Affiliations

  1. National Academy of Science, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

  2. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

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Kapustyan, O.V., Kasyanov, P.O., Valero, J., Zgurovsky, M.Z. (2014). Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System. In: Zgurovsky, M., Sadovnichiy, V. (eds) Continuous and Distributed Systems. Solid Mechanics and Its Applications, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-03146-0_12

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  • DOI: https://doi.org/10.1007/978-3-319-03146-0_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03145-3

  • Online ISBN: 978-3-319-03146-0

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