Pulsating solutions for multidimensional bistable and multistable equations

  • Thomas Giletti
  • Luca RossiEmail author


We investigate the existence of pulsating front-like solutions for spatially periodic heterogeneous reaction–diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.



This study was funded by European Research Council (No. 321186) and Agence Nationale de la Recherche (No. ANR-14-CE25-0013).


  1. 1.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55(8), 949–1032 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math. 65(5), 592–648 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model. I. Species persistence. J. Math. Biol. 51(1), 75–113 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dancer, E.N., Hess, P.: Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math. 419, 125–139 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ducrot, A.: A multi-dimensional bistable nonlinear diffusion equation in a periodic medium. Math. Ann. 366, 783–818 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ducrot, A., Giletti, T., Matano, H.: Existence and convergence to a propagating terrace in one-dimensional reaction–diffusion equations. Trans. Am. Math. Soc. 366(10), 5541–5566 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ding, W., Hamel, F., Zhao, X.-Q.: Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat. Indiana Univ. Math. J. 66(4), 1189–1265 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fang, J., Zhao, X.-Q.: Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc. 17(9), 2243–2288 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fife, P.C., McLeod, J.: The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giletti, T., Matano, H.: Existence and uniqueness of propagating terraces. Commun. Contemp. Math. (to appear) Google Scholar
  12. 12.
    Hamel, F.: Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. J. Math. Pures Appl. 89, 355–399 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer, Berlin (1976)Google Scholar
  14. 14.
    Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196(1), 82–98 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matano, H.: Existence of nontrivial unstable sets for equilibriums of strongly ordered-preserving systems. J. Fac. Sci. Univ. Kyoto 30, 645–673 (1984)zbMATHGoogle Scholar
  16. 16.
    Nadin, G.: The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator. SIAM J. Math. Anal. 4, 2388–2406 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Poláčik, P.: Propagating terraces and the dynamics of front-like solutions of reaction–diffusion equations on \(\mathbb{R}\). Mem. Am. Math. Soc. (to appear) Google Scholar
  18. 18.
    Risler, E.: Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 381–424 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45(6), 511–548 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Xin, J.: Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Differ. Equ. 3, 541–573 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Univ. Lorraine, IECL UMR 7502Vandoeuvre-lès-NancyFrance
  2. 2.CNRS, EHESS, CAMSParisFrance

Personalised recommendations