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Transition fronts in unbounded domains with multiple branches

Abstract

This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts with positive propagating speeds in extended branches, we prove the existence of entire solutions emanating from some almost-planar fronts in some branches. Then, we get that these entire solutions converge to almost-planar fronts in some of the rest branches as time increases if no blocking occurs in these branches. Finally, provided by the complete propagation of every front-like solution emanating from one almost-planar front in every branch, we prove that there is only one type of transition fronts, that is, the entire solutions emanating from some almost-planar fronts in some branches and converging to almost-planar fronts in the rest branches.

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Fig. 1

Notes

  1. A bounded set K is called star-shaped if there is x in the interior \(\mathrm {Int}(K)\) of K such that \(x+t(y-x)\in \mathrm {Int}(K)\) for all \(y\in \partial K\) and \(t\in [0,1)\). Then, we say that K is star-shaped with respect to the point x.

References

  1. Berestycki, H., Bouhours, J., Chapuisat, G.: Front blocking and propagation in cylinders with varying cross section. Calc. Var. Part. Diff. Equ. 55, 1–32 (2016)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  3. 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. 446, 101-123 (2007)

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

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

  6. Berestycki, H., Nirenberg, L.: Traveling fronts in cylinders. Ann. Inst. H. Poincaré, Anal. Non Linéaire 9, 497–572 (1992)

    MathSciNet  Article  Google Scholar 

  7. Chapuisat, G., Grenier, E.: Existence and non-existence of progressive wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased. Commun. Part. Diff. Equ. 30, 1805–1816 (2005)

    Article  Google Scholar 

  8. Ding, W., Hamel, F., Zhao, X.: Propagation phenomena for periodic bistable reaction-diffusion equations. Calc. Var. Part. Diff. Equ. 54, 2517–2551 (2015)

    Article  Google Scholar 

  9. 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)

    MathSciNet  Article  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 periodic medium. Math. Ann. 366, 783–818 (2016)

    MathSciNet  Article  Google Scholar 

  12. Eberle, S.: Front blocking versus propagation in the presence of drift disturbance in the direction of propagation, preprint

  13. 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  Google Scholar 

  14. Fisher, R.A.: The advance of advantageous genes. Ann. Eugenics 7, 335–369 (1937)

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  16. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    Book  Google Scholar 

  17. Guo, H., Hamel, F., Sheng, W.J.: On the mean speed of bistable transition fronts in unbounded domains, J. Math. Pures Appl., to appear

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

    Article  Google Scholar 

  19. 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)

    MathSciNet  Article  Google Scholar 

  20. 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 

  21. Haragus, M., Scheel, A.: Corner defects in almost planar interface propagation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 23, 283–329 (2006)

    MathSciNet  Article  Google Scholar 

  22. Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moskow Univ. Math. Bull. 1, 1–25 (1937)

    Google Scholar 

  23. Matano, H., Nakamura, K.I., Lou, B.: Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogeneity limit. Netw. Heterog. Media 1, 537–568 (2006)

    MathSciNet  Article  Google Scholar 

  24. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    Book  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  27. Pauthier, A.: Entire solution in cylinder-like domains for a bistable reaction-diffusion equation. J. Dyn. Diff. Equ. 30, 1273 (2018)

    MathSciNet  Article  Google Scholar 

  28. Roques, L., Roques, A., Berestycki, H., Kretzschmar, A.: A population facing climate change: joint influences of Allee effects and environmental boundary geometry. Pop. Ecol. 50, 215–225 (2008)

    Article  Google Scholar 

  29. Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution. Oxford University Press, Oxford (1997)

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  37. Xin, J.X.: Analysis and modeling of front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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Acknowledgements

Research partially supported by National Science Foundation (grant no. NSF1826801).

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Correspondence to Hongjun Guo.

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Communicated by P. Rabinowitz.

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Guo, H. Transition fronts in unbounded domains with multiple branches. Calc. Var. 59, 160 (2020). https://doi.org/10.1007/s00526-020-01825-2

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  • DOI: https://doi.org/10.1007/s00526-020-01825-2

Mathematics Subject Classification

  • 35B51
  • 35J61
  • 35K57