Abstract
This paper is concerned with combustion transition fronts in \(\mathbb {R}^{N}\)\((N\ge 1)\). Firstly, we prove the existence and the uniqueness of the global mean speed which is independent of the shape of the level sets of the fronts. Secondly, we show that the planar fronts can be characterized in the more general class of almost-planar fronts. Thirdly, we show the existence of new types of transitions fronts in \(\mathbb {R}^{N}\) which are not standard traveling fronts. Finally, we prove that all transition fronts are monotone increasing in time, whatever shape their level sets may have.
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Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion and nerve propagation. In: Lecture Notes in Math. Partial Differential Equations and Related Topics, vol. 446, Springer, New York, pp. 5–49 (1975)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki, H., Hamel, F.: Generalized travelling waves for reaction–diffusion equations. In: Honor of H. Brezis, Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, vol. 446. Amer. Math. Soc., pp. 101–123 (2007)
Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math. 65, 592–648 (2012)
Berestycki, H., Nicolaenko, B., Scheurer, B.: Traveling waves solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16, 1207–1242 (1985)
Brazhnik, P.K.: Exact solutions for the kinematic model of autowaves in two-dimensional excitable media. Physica D 94, 205–220 (1996)
Bu, Z.-H., Wang, Z.-C.: Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I. Discrete Contin. Dyn. Syst. 37, 2395–2430 (2017)
Bu, Z.-H., Wang, Z.-C.: Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Commun. Pure Appl. Anal. 15, 139–160 (2016)
Fife, P.C., McLeod, J.B.: The approach of solutions of non-linear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
Guo, H., Hamel, F.: Monotonicity of bistable transition fronts in \(\mathbb{R}^N\). J. Elliptic Parabol. Equ. 2, 145–155 (2016)
Hamel, F.: Bistable transition fronts in \(\mathbb{R}^{N}\). Adv. Math. 289, 279–344 (2016)
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)
Hamel, F., Monneau, R., Roquejoffre, J.-M.: Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin. Dyn. Syst. 13, 1069–1096 (2005)
Hamel, F., Monneau, R., Roquejoffre, J.-M.: Asymptotic properties and classification of bistable fronts with Lipschitz level sets. Discrete Contin. Dyn. Syst. 14, 75–92 (2006)
Hamel, F., Rossi, L.: Admissible speeds of transition fronts for non-autonomous monostable equations. SIAM J. Math. Anal. 47, 3342–3392 (2015)
Hamel, F., Roques, L.: Transition fronts for the Fisher-KPP equation. Trans. Am. Math. Soc. 368, 8675–8713 (2016)
Kanel’, JaI: Stabilization of solution of the Cauchy problem for equations encountered in combustion theory. Mat. Sb. 59, 245–288 (1962)
Ma, S., Zhao, X.-Q.: Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete Contin. Dyn. Syst. 21, 259–275 (2008)
Mellet, A., Nolen, J., Roquejoffre, J.-M., Ryzhik, L.: Stability of generalized transition fronts. Commun. Partial Differ. Equ. 34, 521–552 (2009)
Mellet, A., Roquejoffre, J.-M., Sire, Y.: Grneralized fronts for one-dimensional reaction–diffusion equations. Discrete Contin. Dyn. Syst. 26, 303–312 (2010)
Nadin, G.: Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 32, 841–873 (2015)
Nadin, G., Rossi, L.: Propagation phenomena for time heterogeneous KPP reaction–diffusion equations. J. Math. Pures Appl. 98, 633–653 (2012)
Nadin, G., Rossi, L.: Transition waves for Fisher-KPP equations with general time-heterogeneous and space periodic coefficients. Anal. PDE 8, 1351–1377 (2015)
Ninomiya, H., Taniguchi, M.: Existence and global stability of traveling curved fronts in the Allen–Cahn equations. J. Differ. Equ. 213, 204–233 (2005)
Ninomiya, H., Taniguchi, M.: Global stability of traveling curved fronts in the Allen–Cahn equations. Discrete Contin. Dyn. Syst. 15, 819–832 (2006)
Ninomiya, H., Taniguchi, M.: Stability of traveling curved fronts in a curvature flow with driving force. Methods Appl. Anal. 8, 429–450 (2001)
Ninomiya, H., Taniguchi, M.: Traveling curved fronts of a mean curvature flow with constant driving force. In: Free Boundary Problems: Theory and Applications I. GAKUTO International Series. Mathematical Sciences and Applications, vol. 13, pp. 206–221 (2000)
Nolen, J., Roquejoffre, J.-M., Ryzhik, L., Zlatos̆, A.: Existence and non-existence of Fisher-KPP transition fronts. Arch. Ration. Mech. Anal. 203, 217–246 (2012)
Nolen, J., Ryzhik, L.: Traveling waves in a one-dimensional heterogeneous medium. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1021–1047 (2009)
Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011–1060 (2004)
Shen, W.: Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations. J. Dyn. Differ. Equ. 23, 1–44 (2011)
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)
Taniguchi, M.: Traveling fronts of pyramidal shapes in the Allen–Cahn equation. SIAM J. Math. Anal. 39, 319–344 (2007)
Taniguchi, M.: The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations. J. Differ. Equ. 246, 2103–2130 (2009)
Taniguchi, M.: Multi-dimensional traveling fronts in bistable reaction–diffusion equations. Discrete Contin. Dyn. Syst. 32, 1011–1046 (2012)
Wang, Z.-C., Bu, Z.-H.: Nonplanar traveling fronts in reaction–diffusion equations with combustion and degenerate Fisher-KPP nonlinearities. J. Differ. Equ. 260, 6405–6450 (2016)
Zlatos̆, A.: Generalized traveling waves in disordered media: existence, uniqueness, and stability. Arch. Ration. Mech. Anal. 208, 447–480 (2013)
Zlatos̆, A.: Transition fronts in inheomogeneous Fisher-KPP reaction–diffusion equations. J. Math. Pures Appl. 98, 89–102 (2012)
Zlatos̆, A.: Propagation of reactions in inhomogeneous media. Commun. Pure Appl. Math. 70, 884–949 (2017)
Zlatos̆, A.: Existence and non-existence of transition fronts for bistable and ignition reactions. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 1687–1705 (2017)
Acknowledgements
The first author and the second author would like to thank Professor François Hamel of Aix-Marseille University for the valuable discussions. They was supported by the China Scholarship Council. The third author was supported by NNSF of China (11371179, 11731005) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2016-ct12, lzujbky-2017-ct01). This work has been partly carried out in the framework of the ANR DEFI Project NONLOCAL (ANR-14-CE25-0013), of Archimèdes Labex (ANR-11-LABX-0033), of the A*MIDEX Project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR), and of the ERC Project ReaDi - Reaction–Diffusion Equations, Propagation and Modelling, Grant Agreement n. 321186 funded by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013).
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Bu, ZH., Guo, H. & Wang, ZC. Transition Fronts of Combustion Reaction Diffusion Equations in \(\mathbb {R}^{N}\). J Dyn Diff Equat 31, 1987–2015 (2019). https://doi.org/10.1007/s10884-018-9675-x
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DOI: https://doi.org/10.1007/s10884-018-9675-x