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Recent Advances in Reaction-Diffusion Equations with Non-ideal Relays

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Control of Self-Organizing Nonlinear Systems

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism—rattling, which indicates how one should reset the continuous model to make it well posed.

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Acknowledgments

The authors are grateful for the support of the DFG project SFB 910 and the DAAD project G-RISC. The work of the first author was partially supported by the Berlin Mathematical School. The work of the second author was partially supported by the DFG Heisenberg programme. The work of the third author was partially supported by Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026, JSC “Gazprom neft”, by the Saint-Petersburg State University research grant 6.38.223.2014 and RFBR 15-01-03797a.

Author information

Authors and Affiliations

  1. Institute of Mathematics I, Free University of Berlin, Arnimallee 7, 14195, Berlin, Germany

    Mark Curran

  2. Institute of Mathematics I, Free University of Berlin, Arnimallee 3, 14195, Berlin, Germany

    Pavel Gurevich

  3. Peoples’ Friendship University of Russia, Miklukho-Maklaya Str. 6, 117198, Moscow, Russia

    Pavel Gurevich

  4. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Sergey Tikhomirov

  5. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, 199178, Saint Petersburg, Russia

    Sergey Tikhomirov

Authors
  1. Mark Curran
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  2. Pavel Gurevich
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  3. Sergey Tikhomirov
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Corresponding author

Correspondence to Mark Curran .

Editor information

Editors and Affiliations

  1. Inst für Theoretische Physik, Technische Universität Berlin, Berlin, Berlin, Germany

    Eckehard Schöll

  2. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Sabine H. L. Klapp

  3. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Philipp Hövel

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Curran, M., Gurevich, P., Tikhomirov, S. (2016). Recent Advances in Reaction-Diffusion Equations with Non-ideal Relays. In: Schöll, E., Klapp, S., Hövel, P. (eds) Control of Self-Organizing Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-28028-8_11

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