Skip to main content

Monotonicity of Bistable Transition Fronts in ℝN

Abstract

This paper is concerned with the monotonicity of transition fronts for bistable reaction diffusion equations. Transition fronts generalize the standard notions of traveling fronts. Known examples of standard traveling fronts are the planar fronts and the fronts with conical-shaped or pyramidal level sets which are invariant in a moving frame. Other more general non-standard transition fronts with more complex level sets were constructed recently. In this paper, we prove the time monotonicity of all bistable transition fronts with non-zero global mean speed, whatever shape their level sets may have.

This is a preview of subscription content, access via your institution.

References

  1. D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33–76.

    MathSciNet  Article  MATH  Google Scholar 

  2. H. Berestycki, F. Hamel, Generalized traveling waves for reaction-diffusion equations, In: Perspectives in Nonlinear Partial Differential Equations. In honor of H. Brezis, Amer. Math. Soc., Contemp. Math. 446, 2007, 101–123.

    Article  MATH  Google Scholar 

  3. H. Berestycki, F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), 592–648.

    MathSciNet  Article  MATH  Google Scholar 

  4. X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya, J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré, Analyse Non Linéaire 24 (2007), 369–393.

    MathSciNet  Article  MATH  Google Scholar 

  5. M. Del Pino, M. Kowalczyk, J. Wei, Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math. 66 (2013), 481–547.

    MathSciNet  Article  MATH  Google Scholar 

  6. P.C. Fife, J.B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal. 65 (1977), 335–361.

    MathSciNet  Article  MATH  Google Scholar 

  7. F. Hamel, Bistable transition fronts inN, Adv. Math. 289 (2016), 279–344.

    MathSciNet  Article  MATH  Google Scholar 

  8. F. Hamel, R. Monneau, Solutions of semilinear elliptic equations inNwith conical-shaped level sets, Comm. Part. Diff. Equations 25 (2000), 769–819.

  9. F. Hamel, R. Monneau, J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Disc. Cont. Dyn. Syst. A 13 (2005), 1069–1096.

    MathSciNet  Article  MATH  Google Scholar 

  10. F. Hamel, R. Monneau, J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Syst. A 14 (2006), 75–92.

    MathSciNet  MATH  Google Scholar 

  11. F. Hamel, L. Rossi, Admissible speeds of transition fronts for non-autonomous monostable equations, SIAM J. Math. Anal. 47 (2015), 3342–3392.

    MathSciNet  Article  MATH  Google Scholar 

  12. F. Hamel, L. Rossi, Transition fronts for the Fisher-KPP equation, Trans. Amer. Math. Soc. 368 (2016), 8675–8713.

    MathSciNet  Article  MATH  Google Scholar 

  13. A. Mellet, J. Nolen, J.-M. Roquejoffre, L. Ryzhik, Stability of generalized transition fronts, Comm. Part. Diff. Equations 34 (2009), 521–552.

    MathSciNet  Article  MATH  Google Scholar 

  14. A. Mellet, J.-M. Roquejoffre, Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Disc. Cont. Dyn. Syst. A 26 (2010), 303–312.

    MathSciNet  MATH  Google Scholar 

  15. G. Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré, Non Linear Anal. 32 (2015), 841–873.

    MathSciNet  Article  MATH  Google Scholar 

  16. G. Nadin, L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl. 98 (2012), 633–653.

    MathSciNet  Article  MATH  Google Scholar 

  17. G. Nadin, L. Rossi, Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coeffcients, Anal. PDE 8 (2015), 1351–1377.

    MathSciNet  Article  MATH  Google Scholar 

  18. H. Ninomiya, M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Equations 213 (2005), 204–233.

    MathSciNet  Article  MATH  Google Scholar 

  19. H. Ninomiya, M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Cont. Dyn. Syst. A 15 (2006), 819–832.

    MathSciNet  Article  MATH  Google Scholar 

  20. J. Nolen, J.-M. Roquejoffre, L. Ryzhik, A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal. 203 (2012), 217–246.

    MathSciNet  Article  MATH  Google Scholar 

  21. J. Nolen, L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré, Analyse Non Linéaire 26 (2009), 1021–1047.

    MathSciNet  Article  MATH  Google Scholar 

  22. J.-M. Roquejoffre, V. Roussier-Michon, Nontrivial large-time behavior in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. 188 (2009), 207–233.

    MathSciNet  Article  MATH  Google Scholar 

  23. W. Shen, Traveling waves in diffusive random media, J. Dyn. Diff. Equations 16 (2004), 1011–1060.

    MathSciNet  Article  MATH  Google Scholar 

  24. W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dyn. Diff. Equations 23 (2011), 1–44.

    MathSciNet  Article  MATH  Google Scholar 

  25. W. Shen, Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, preprint http://arxiv.org/abs/1408.3848 url).

  26. M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equation, SIAM J. Math. Anal. 39 (2007), 319–344.

    MathSciNet  Article  MATH  Google Scholar 

  27. M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Diff. Equations 246 (2009), 2103–2130.

    MathSciNet  Article  MATH  Google Scholar 

  28. M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Disc. Cont. Dyn. Syst. A 32 (2012), 1011–1046.

    MathSciNet  Article  MATH  Google Scholar 

  29. A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. 98 (2012), 89–102.

    MathSciNet  Article  MATH  Google Scholar 

  30. A. Zlatoš, Generalized traveling waves in disordered media: existence, uniqueness, and stability, Arch. Ration. Mech. Anal. 208 (2013), 447–480.

    MathSciNet  Article  MATH  Google Scholar 

  31. A. Zlatoš, Existence and non-existence of transition fronts for bistable and ignition reactions, Ann. Inst. H. Poincaré, Analyse Non Linéaire, forthcoming.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hongjun Guo.

Additional information

Dedicated to Professor David Kinderlehrer

This work has been carried out thanks to the support of the A*MIDEX project (no ANR-11-IDEX-0001-02) and Archimède Labex (no ANR-11-LABX-0033) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). The research leading to these results has also received funding from the ANR within the project NONLOCAL (no ANR-14-CE25-0013) and from the European Research Council under the European Unions Seventh Framework Programme(FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi - Reaction- Diffusion Equations, Propagation and Modelling.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, H., Hamel, F. Monotonicity of Bistable Transition Fronts in ℝN. J Elliptic Parabol Equ 2, 145–155 (2016). https://doi.org/10.1007/BF03377398

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03377398

2010 Mathematics Subject Classication

  • Primary: 35B08, 35C07, 35K57
  • Secondary: 35B45

Key words and phrases

  • Reaction-diffusion equations
  • Transition fronts
  • Monotonicity