Traveling waves with paraboloid like interfaces for balanced bistable dynamics
Chen, Xinfu ; Guo, Jong-Shenq ; Hamel, François ; Ninomiya, Hirokazu ; Roquejoffre, Jean-Michel
Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007), p. 369-393 / Harvested from Numdam
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     author = {Chen, Xinfu and Guo, Jong-Shenq and Hamel, Fran\c cois and Ninomiya, Hirokazu and Roquejoffre, Jean-Michel},
     title = {Traveling waves with paraboloid like interfaces for balanced bistable dynamics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {24},
     year = {2007},
     pages = {369-393},
     doi = {10.1016/j.anihpc.2006.03.012},
     mrnumber = {2319939},
     zbl = {1132.35396},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2007__24_3_369_0}
}
Chen, Xinfu; Guo, Jong-Shenq; Hamel, François; Ninomiya, Hirokazu; Roquejoffre, Jean-Michel. Traveling waves with paraboloid like interfaces for balanced bistable dynamics. Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) pp. 369-393. doi : 10.1016/j.anihpc.2006.03.012. http://gdmltest.u-ga.fr/item/AIHPC_2007__24_3_369_0/

[1] Alberti G., Ambrosio L., Cabré X., On a long-standing conjecture of E. De Giorgi: old and recent results, Acta Appl. Math. 65 (2001) 9-33. | MR 1843784 | Zbl 1121.35312

[2] Allen S., Cahn J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall. 27 (1979) 1084-1095.

[3] Ambrosio L., Cabré X., Entire solutions of semilinear elliptic equations in R 3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000) 725-739. | MR 1775735 | Zbl 0968.35041

[4] Aronson D.G., Weinberger H.F., Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978) 33-76. | MR 511740 | Zbl 0407.92014

[5] Berestycki H., Caffarelli L., Nirenberg L., Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 (1997) 1089-1111. | MR 1470317 | Zbl 0906.35035

[6] Berestycki H., Hamel F., Monneau R., One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000) 375-396. | MR 1763653 | Zbl 0954.35056

[7] H. Berestycki, B. Larrouturou, Planar travelling front solutions of reaction-diffusion problems, preprint.

[8] Bonnet A., Hamel F., Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal. 31 (1999) 80-118. | MR 1742304 | Zbl 0942.35072

[9] Caffarelli L.A., Cabré X., Fully Nonlinear Elliptic Equations, Colloquium Publications, vol. 43, Amer. Math. Soc., 1995. | MR 1351007 | Zbl 0834.35002

[10] Carr J., Pego R.L., Invariant manifolds for metastable patterns in u t =ϵ 2 u xx -fu, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990) 133-160. | MR 1076358 | Zbl 0738.35023

[11] Chen X., Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations 96 (1992) 116-141. | Zbl 0765.35024

[12] Chen X., Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations 19 (1994) 1371-1395. | Zbl 0811.35098

[13] Chen X., Generation, propagation, and annihilation of metastable patterns, J. Differential Equations 206 (2004) 399-437. | MR 2095820 | Zbl 1061.35014

[14] X. Chen, J.-S. Guo, H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, in press. | Zbl 1123.35024

[15] Chen X., Taniguchi M., Instability of spherical interfaces in a nonlinear free boundary problem, Adv. Differential Equations 5 (2000) 747-772. | MR 1750117 | Zbl 1012.35081

[16] De Giorgi E., Convergence problems for functionals and operators, in: Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, 1979, pp. 131-188. | MR 533166 | Zbl 0405.49001

[17] De Mottoni P., Schatzman M., Development of interfaces in R N , Proc. Roy. Soc. Edinburgh Sect. A 116 (1990) 207-220. | MR 1084732 | Zbl 0725.35009

[18] Ei S.-I., The motion of weakly interaction pulses in reaction-diffusion systems, J. Dynamics Differential Equations 14 (2002) 85-137. | Zbl 1007.35039

[19] Evans L.C., Soner H.M., Souganidis P.E., Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992) 1097-1123. | MR 1177477 | Zbl 0801.35045

[20] Fife P.C., Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 53, 1988. | MR 981594 | Zbl 0684.35001

[21] Fife P.C., Mcleod J.B., The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal. 65 (1977) 335-361. | MR 442480 | Zbl 0361.35035

[22] Fusco G., A geometric approach to the dynamics of u t =ϵ 2 u xx +fu for small ε, in: Kirchgassner (Ed.), Lecture Notes in Physics, vol. 359, 1990, pp. 53-73. | Zbl 0715.35038

[23] Fusco G., Hale J.K., Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynamics Differential Equations 1 (1989) 75-94. | MR 1010961 | Zbl 0684.34055

[24] Ghoussoub N., Gui C., On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998) 481-491. | MR 1637919 | Zbl 0918.35046

[25] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1997. | Zbl 0562.35001

[26] Hamel F., Monneau R., Solutions of semilinear elliptic equations in R N with conical-shaped level sets, Comm. Partial Differential Equations 25 (2000) 769-819. | MR 1759793 | Zbl 0952.35041

[27] Hamel F., Monneau R., Existence and uniqueness for a free boundary problem arising in combustion theory, Interfaces Free Boundaries 4 (2002) 167-210. | MR 1950528 | Zbl 1078.80004

[28] Hamel F., Monneau R., Roquejoffre J.-M., Stability of traveling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. 37 (2004) 469-506. | Numdam | MR 2060484 | Zbl 1085.35075

[29] Hamel F., Monneau R., Roquejoffre J.-M., Existence and qualitative properties of multidimensional conical bistable fronts, Disc. Cont. Dyn. Systems 13 (2005) 1069-1096. | MR 2166719 | Zbl 1097.35078

[30] Hamel F., Monneau R., Roquejoffre J.-M., Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Systems 14 (2006) 75-92. | MR 2170314 | Zbl pre05125745

[31] Hamel F., Nadirashvili N., Travelling waves and entire solutions of the Fisher-KPP equation in R N , Arch. Rational Mech. Anal. 157 (2001) 91-163. | MR 1830037 | Zbl 0987.35072

[32] Haragus M., Scheel A., Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 283-329. | Numdam | MR 2217654 | Zbl 1098.35085

[33] Ilmanen T., Convergence of the Allen-Cahn equation to Brakke's motion, J. Differential Geom. 38 (1993) 417-461. | Zbl 0784.53035

[34] 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 Internat. Ser. Math. Sci. Appl., vol. 13, 2000, pp. 206-221. | MR 1793036 | Zbl 0957.35124

[35] Ninomiya H., Taniguchi M., Stability of traveling curved fronts in a curvature flow with driving force, Methods Appl. Anal. 8 (2001) 429-450. | MR 1904754 | Zbl 1007.35004

[36] Ninomiya H., Taniguchi M., Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations 213 (2005) 204-233. | Zbl pre02182113

[37] H. Ninomiya, M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Cont. Dyn. Systems, submitted for publication. | Zbl 1118.35012

[38] Ouyang T., Shi J., Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations 146 (1998) 121-156. | MR 1625731 | Zbl 0918.35049

[39] O. Savin, Phase transitions: regularity of flat level sets, preprint.

[40] Soner H.M., Motion of a set by the curvature of its boundary, J. Differential Equations 101 (1993) 313-372. | MR 1204331 | Zbl 0769.35070