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

  • Mark CurranEmail author
  • Pavel Gurevich
  • Sergey Tikhomirov
Part of the Understanding Complex Systems book series (UCS)


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.


Initial Data Free Boundary Spatial Profile Monotonicity Formula Scalar Input 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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 and RFBR 15-01-03797a.


  1. 1.
    A. Visintin, Differential Models of Hysteresis. Applied Mathematical Sciences (Springer-Verglag, Berlin, 1994)Google Scholar
  2. 2.
    P. Krejčí, Hysteresis Convexity and Dissipation in Hyperbolic Equations. GAKUTO International series (Gattötoscho, 1996)Google Scholar
  3. 3.
    M. Brokate, J. Sprekels, Hysteresis and Phase Transitions. Applied Mathematical Sciences (Springer-Verlag, New York, 1996)Google Scholar
  4. 4.
    A. Visintin, Acta Applicandae Mathematicae 132(1), 635 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Visintin, Discrete Contin. Dyn. Syst., Ser. S 8(4), 793 (2015)Google Scholar
  6. 6.
    F. Hoppensteadt, W. Jäger, in Biological Growth and Spread. Lecture Notes in Biomathematics, vol. 38, ed. by W. Jäger, H. Rost, P. Tautu (Springer, Berlin Heidelberg, 1980), pp. 68–81Google Scholar
  7. 7.
    F. Hoppensteadt, W. Jäger, C. Pöppe, in Modelling of Patterns, in Space and Time. Lecture Notes in Biomathematics, ed. by W. Jäger, J.D. Murray (Springer, Berlin Heidelberg, 1984), vol. 55, pp. 123–134Google Scholar
  8. 8.
    A. Marciniak-Czochra, Math. Biosci. 199(1), 97 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Köthe, Hysteresis-driven pattern formation in reaction-diffusion-ode models. Ph.D. thesis, University of Heidelberg (2013)Google Scholar
  10. 10.
    M. Krasnosel’skii, M. Niezgodka, A. Pokrovskii, Systems with Hysteresis (Springer, Berlin, 2012)Google Scholar
  11. 11.
    P. Gurevich, S. Tikhomirov, R. Shamin, SIAM J. Math. Anal. 45(3), 1328 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    H.W. Alt, Control Cybern. 14(1–3), 171 (1985)MathSciNetGoogle Scholar
  13. 13.
    A. Visintin, SIAM J. Math. Anal. 17(5) (1986)Google Scholar
  14. 14.
    T. Aiki, J. Kopfová, in Recent Advances in Nonlinear Analysis (2008), pp. 1–10Google Scholar
  15. 15.
    P. Krejčí, J. Physics.: Conf. Ser. (22), 103 (2005)Google Scholar
  16. 16.
    E. Mischenko, N. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, New York, 1980)CrossRefGoogle Scholar
  17. 17.
    C. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences, vol. 191 (Springer International Publishing, 2015)Google Scholar
  18. 18.
    D. Apushkinskaya, N. Uraltseva, St. Petersbg. Math. J. 25(2), 195 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H. Shahgholian, N. Uraltseva, G.S. Weiss, Adv. Math. 221(3), 861 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Gurevich, S. Tikhomirov, Nonlinear Anal. 75(18), 6610 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    P. Gurevich, S. Tikhomirov, Mathematica Bohemica (Proc. Equadiff 2013) 139(2), 239 (2014)Google Scholar
  22. 22.
    M. Curran, Local well-poseness of a reaction-diffusion equation with hysteresis. Master’s thesis, Fachbereich Mathematik und Informatik, Freie Universität Berlin (2014)Google Scholar
  23. 23.
    D. Apushkinskaya, N. Uraltseva, Interfaces and Free Boundaries 17(1), 93 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    P. Gurevich, S. Tikhomirov, arXiv:1504.02385 [math.AP] (2015)
  25. 25.
    O. Ladyzhenskaya, V. Solonnikov, N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type (American Mathematical Society, Providence, Rohde Island, 1968)Google Scholar
  26. 26.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Carnegie-Rochester Conference Series on Public Policy (North-Holland Publishing Company, 1978)Google Scholar
  27. 27.
    S. Ivasishen, Math. USSR-Sb (4), 461 (1981)Google Scholar
  28. 28.
    L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics (American Mathematical Soc., 2005)Google Scholar
  29. 29.
    D. Apushkinskaya, H. Shahgholian, N. Uraltseva, J. Math. Sci. 115(6), 2720 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    P. Gurevich, arXiv:1504.02673 [math.AP] (2015)

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mark Curran
    • 1
    Email author
  • Pavel Gurevich
    • 2
    • 3
  • Sergey Tikhomirov
    • 4
    • 5
  1. 1.Institute of Mathematics IFree University of BerlinBerlinGermany
  2. 2.Institute of Mathematics IFree University of BerlinBerlinGermany
  3. 3.Peoples’ Friendship University of RussiaMoscowRussia
  4. 4.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  5. 5.Chebyshev LaboratorySt. Petersburg State UniversitySaint PetersburgRussia

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