Transition fronts of combustion reaction diffusion equations in \(\mathbb{R}^N\). (English) Zbl 1423.35161
Summary: This paper is concerned with combustion transition fronts in \(\mathbb{R}^N(N \geq 1)\). 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.
MSC:
| 35K40 | Second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 35C07 | Traveling wave solutions |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
reaction-diffusion equations; combustion nonlinearity; transition front; qualitative propertiesReferences:
| [1] | 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) · Zbl 0325.35050 |
| [2] | Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33-76 (1978) · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5 |
| [3] | 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) · Zbl 1200.35169 |
| [4] | Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math. 65, 592-648 (2012) · Zbl 1248.35039 · doi:10.1002/cpa.21389 |
| [5] | Berestycki, H., Nicolaenko, B., Scheurer, B.: Traveling waves solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16, 1207-1242 (1985) · Zbl 0596.76096 · doi:10.1137/0516088 |
| [6] | Brazhnik, P.K.: Exact solutions for the kinematic model of autowaves in two-dimensional excitable media. Physica D 94, 205-220 (1996) · Zbl 0890.58088 · doi:10.1016/0167-2789(96)00042-5 |
| [7] | 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) · Zbl 1358.35014 · doi:10.3934/dcds.2017104 |
| [8] | 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) · Zbl 1330.35210 · doi:10.3934/cpaa.2016.15.139 |
| [9] | 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) · Zbl 0361.35035 · doi:10.1007/BF00250432 |
| [10] | Guo, H., Hamel, F.: Monotonicity of bistable transition fronts in \[\mathbb{R}^N\] RN. J. Elliptic Parabol. Equ. 2, 145-155 (2016) · Zbl 1386.35009 · doi:10.1007/BF03377398 |
| [11] | Hamel, F.: Bistable transition fronts in \[\mathbb{R}^N\] RN. Adv. Math. 289, 279-344 (2016) · Zbl 1348.35111 · doi:10.1016/j.aim.2015.11.033 |
| [12] | Hamel, F., Monneau, R.: Solutions of semilinear elliptic equations in \[\mathbb{R}^N\] RN with conical-shaped level sets. Commun. Partial Differ. Equ. 25, 769-819 (2000) · Zbl 0952.35041 · doi:10.1080/03605300008821532 |
| [13] | Hamel, F., Monneau, R., Roquejoffre, J.-M.: Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin. Dyn. Syst. 13, 1069-1096 (2005) · Zbl 1097.35078 · doi:10.3934/dcds.2005.13.1069 |
| [14] | 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) · Zbl 1194.35151 |
| [15] | Hamel, F., Rossi, L.: Admissible speeds of transition fronts for non-autonomous monostable equations. SIAM J. Math. Anal. 47, 3342-3392 (2015) · Zbl 1334.35118 · doi:10.1137/140995519 |
| [16] | Hamel, F., Roques, L.: Transition fronts for the Fisher-KPP equation. Trans. Am. Math. Soc. 368, 8675-8713 (2016) · Zbl 1362.35157 · doi:10.1090/tran/6609 |
| [17] | Kanel’, JaI: Stabilization of solution of the Cauchy problem for equations encountered in combustion theory. Mat. Sb. 59, 245-288 (1962) |
| [18] | Ma, S., Zhao, X.-Q.: Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete Contin. Dyn. Syst. 21, 259-275 (2008) · Zbl 1149.35320 · doi:10.3934/dcds.2008.21.259 |
| [19] | Mellet, A., Nolen, J., Roquejoffre, J.-M., Ryzhik, L.: Stability of generalized transition fronts. Commun. Partial Differ. Equ. 34, 521-552 (2009) · Zbl 1173.35021 · doi:10.1080/03605300902768677 |
| [20] | Mellet, A., Roquejoffre, J.-M., Sire, Y.: Grneralized fronts for one-dimensional reaction-diffusion equations. Discrete Contin. Dyn. Syst. 26, 303-312 (2010) · Zbl 1180.35294 |
| [21] | Nadin, G.: Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 32, 841-873 (2015) · Zbl 1364.35146 · doi:10.1016/j.anihpc.2014.03.007 |
| [22] | Nadin, G., Rossi, L.: Propagation phenomena for time heterogeneous KPP reaction-diffusion equations. J. Math. Pures Appl. 98, 633-653 (2012) · Zbl 1388.35101 · doi:10.1016/j.matpur.2012.05.005 |
| [23] | Nadin, G., Rossi, L.: Transition waves for Fisher-KPP equations with general time-heterogeneous and space periodic coefficients. Anal. PDE 8, 1351-1377 (2015) · Zbl 1351.35014 · doi:10.2140/apde.2015.8.1351 |
| [24] | Ninomiya, H., Taniguchi, M.: Existence and global stability of traveling curved fronts in the Allen-Cahn equations. J. Differ. Equ. 213, 204-233 (2005) · Zbl 1159.35378 · doi:10.1016/j.jde.2004.06.011 |
| [25] | Ninomiya, H., Taniguchi, M.: Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete Contin. Dyn. Syst. 15, 819-832 (2006) · Zbl 1118.35012 · doi:10.3934/dcds.2006.15.819 |
| [26] | Ninomiya, H., Taniguchi, M.: Stability of traveling curved fronts in a curvature flow with driving force. Methods Appl. Anal. 8, 429-450 (2001) · Zbl 1007.35004 |
| [27] | 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) · Zbl 0957.35124 |
| [28] | 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) · Zbl 1267.35108 · doi:10.1007/s00205-011-0449-4 |
| [29] | Nolen, J., Ryzhik, L.: Traveling waves in a one-dimensional heterogeneous medium. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1021-1047 (2009) · Zbl 1178.35205 · doi:10.1016/j.anihpc.2009.02.003 |
| [30] | Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011-1060 (2004) · Zbl 1082.35081 · doi:10.1007/s10884-004-7832-x |
| [31] | Shen, W.: Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations. J. Dyn. Differ. Equ. 23, 1-44 (2011) · Zbl 1223.35103 · doi:10.1007/s10884-010-9200-3 |
| [32] | 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) · Zbl 1357.35079 · doi:10.1090/tran/6726 |
| [33] | Taniguchi, M.: Traveling fronts of pyramidal shapes in the Allen-Cahn equation. SIAM J. Math. Anal. 39, 319-344 (2007) · Zbl 1143.35073 · doi:10.1137/060661788 |
| [34] | Taniguchi, M.: The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations. J. Differ. Equ. 246, 2103-2130 (2009) · Zbl 1176.35101 · doi:10.1016/j.jde.2008.06.037 |
| [35] | Taniguchi, M.: Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete Contin. Dyn. Syst. 32, 1011-1046 (2012) · Zbl 1235.35067 · doi:10.3934/dcds.2012.32.1011 |
| [36] | 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) · Zbl 1426.35138 · doi:10.1016/j.jde.2015.12.045 |
| [37] | Zlatos̆, A.: Generalized traveling waves in disordered media: existence, uniqueness, and stability. Arch. Ration. Mech. Anal. 208, 447-480 (2013) · Zbl 1270.35173 · doi:10.1007/s00205-012-0600-x |
| [38] | Zlatos̆, A.: Transition fronts in inheomogeneous Fisher-KPP reaction-diffusion equations. J. Math. Pures Appl. 98, 89-102 (2012) · Zbl 1252.35110 · doi:10.1016/j.matpur.2011.11.007 |
| [39] | Zlatos̆, A.: Propagation of reactions in inhomogeneous media. Commun. Pure Appl. Math. 70, 884-949 (2017) · Zbl 1456.35112 · doi:10.1002/cpa.21653 |
| [40] | 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) · Zbl 1379.35165 · doi:10.1016/j.anihpc.2016.11.004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
