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Propagating speeds of bistable transition fronts in spatially periodic media

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Abstract

This paper is concerned with the propagating speeds of transition fronts in \(\mathbb {R}^N\) for spatially periodic bistable reaction–diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the a priori assumption that there exist pulsating fronts for every direction e with nonzero speeds, we show some continuity and differentiability properties of the front speeds and profiles with respect to the direction e. Finally, we prove that the propagating speed of any transition front is larger than the infimum of speeds of pulsating fronts and less than the supremum of speeds of pulsating fronts.

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References

  1. Alfro, M., Giletti, T.: Varying the direction of propagation in reaction–diffusion equations in periodic media. Netw. Heterog. Media 11, 369–393 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Berestycki, H., Hamel, F., Matano, H.: Bistable travelling waves around an obstacle. Commun. Pure Appl. Math. 62, 729–788 (2009)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, W., Hamel, F., Zhao, X.: Bistable pulsating fronts for reaction–diffusion equations in a periodic habitat. Indiana Univ. Math. J. 66, 1189–1265 (2017)

  9. Ding, W., Hamel, F., Zhao, X.: Transition fronts for periodic bistable reaction–diffusion equations. Calc. Var. Part. Differ. Equ. 54, 2517–2551 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ducasse, R., Rossi, L.: Blocking and invasion for reaction–diffusion equations in periodic media, preprint (https://arxiv.org/abs/1711.07389)

  11. Ducrot, A.: A multi-dimensional bistable nonlinear diffusion equation in a periodic medium. Math. Ann. 366, 783–818 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ducrot, A., Giletti, T., Matano, H.: Existence and convergence to a propagating terrace in one-dimensional reaction–diffusion equations. Trans. Am. Math. Soc. 366, 5541–5566 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. El Smaily, M., Hamel, F., Huang, R.: Two-dimensional curved fronts in a periodic shear flow. Nonlinear Anal. 74, 6469–6486 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang, J., Zhao, X.-Q.: Bistable traveling waves for monotone semiflows with applications. J. Europe Math. Soc. 17, 2243–2288 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Hamel, F.: Bistable transition fronts in \(\mathbb{R}^N\). Adv. Math. 289, 279–344 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hamel, F., Monneau, R.: Solutions of semilinear elliptic equations in \(\mathbb{R}^N\) with conical-shaped level sets. Commun. Partial Differ. Equ. 25, 769–819 (2000)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Hamel, F., Rossi, L.: Transition fronts for the Fisher–KPP equation. Trans. Am. Math. Soc. 368, 8675–8713 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mellet, A., Nolen, J., Roquejoffre, J.-M., Ryzhik, L.: Stability of generalized transition fronts. Commun. Partial Differ. Equ. 34, 521–552 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Ninomiya, H., Taniguchi, M.: Existence and global stability of traveling curved fronts in the Allen–Cahn equations. J. Differ. Equ. 213, 204–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Nolen, J., Ryzhik, L.: Traveling waves in a one-dimensional heterogeneous medium. Ann. Inst. H. Poincaré Anal. Nonlinéaire 26, 1021–1047 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011–1060 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, W.: Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations. J. Dyn. Differ. Equ. 23, 1–44 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, W., Shen, Z.: Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type. Trans. Am. Math. Soc. 369, 2573–2613 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Taniguchi, M.: The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations. J. Differ. Equ. 246, 2103–2130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Xin, X.: Existence and uniqueness of travelling waves in a reaction–diffusion equation with combustion nonlinearity. Indiana Univ. Math. J. 40, 985–1008 (1991)

    Article  MathSciNet  Google Scholar 

  40. Xin, X.: Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Differ. Equ. 3, 541–573 (1991)

    Article  MATH  Google Scholar 

  41. Xin, J.X.: Existence of planar flame fronts in convective–diffusive periodic media. Arch. Ration. Mech. Anal. 121, 205–233 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xin, J.X.: Existence and nonexistence of traveling waves and reaction–diffusion front propagation in periodic media. J. Stat. Phys. 73, 893–926 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xin, J.X., Zhu, J.: Quenching and propagation of bistable reaction–diffusion fronts in multidimensional periodic media. Physica D 81, 94–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Zlatoš, A.: Existence and non-existence of transition fronts for bistable and ignition reactions. Ann. Inst. H. Poincaré Anal. Nonlinéaire 34, 1687–1705 (2017)

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Acknowledgements

The author was supported by the China Scholarship Council for 3 years of study at Aix Marseille Université. The author is grateful to Professor François Hamel for his patient discussions and helpful suggestions, and to the anonymous referee for interesting comments which led to an improvement of the article.

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Authors and Affiliations

  1. Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

    Hongjun Guo

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  1. Hongjun Guo
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Correspondence to Hongjun Guo.

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Communicated by P. Rabinowitz.

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Guo, H. Propagating speeds of bistable transition fronts in spatially periodic media. Calc. Var. 57, 47 (2018). https://doi.org/10.1007/s00526-018-1327-9

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