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Heteroclinic connections for multidimensional bistable reaction-diffusion equations
1. | Aix-Marseille Université & Institut Universitaire de France, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France |
2. | Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4 |
References:
[1] |
H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. |
[2] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Amer. Math. Soc., Contemp. Math., 446, (2007), 101-123. |
[3] |
H. Berestycki and F. Hamel, "Reaction-Diffusion Equations and Propagation Phenomena,", Applied Mathematical Sciences, ().
|
[4] |
H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: doi:10.1215/S0012-7094-00-10331-6. |
[5] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
doi: doi:10.1016/j.matpur.2004.10.006. |
[6] |
H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 9 (1992), 497-572. |
[7] |
G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Diff. Equations, 236 (2007), 237-279.
doi: doi:10.1016/j.jde.2007.01.021. |
[8] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 24 (2007), 369-393.
doi: doi:10.1016/j.anihpc.2006.03.012. |
[9] |
M. El Smaily, Min-max formulæ for the speeds of pulsating travelling fronts in periodic excitable media, Ann. Mat. Pura Appl., 189 (2010), 47-66.
doi: doi:10.1007/s10231-009-0100-2. |
[10] |
P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Philadelphia, PA, 1988. |
[11] |
P. C. Fife and J. B. McLeod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.
doi: doi:10.1007/BF00250432. |
[12] |
F. Hamel, Formules min-max pour les vitesses d'ondes progressives multidimensionnelles, Ann. Fac. Sci. Toulouse, 8 (1999), 259-280. |
[13] |
F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets, Comm. Part. Diff. Equations, 25 (2000), 769-819.
doi: doi:10.1080/03605300008821532. |
[14] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Scient. Ecole Norm. Sup., 37 (2004), 469-506. |
[15] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Disc. Cont. Dyn. Systems, 13 (2005), 1069-1096.
doi: doi:10.3934/dcds.2005.13.1069. |
[16] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Systems, 14 (2006), 75-92. |
[17] |
F. Hamel and N. Nadirashvili, Travelling waves and entire solutions of the Fisher-KPP equation in $\R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: doi:10.1007/PL00004238. |
[18] |
M. Haragus, A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 23 (2006), 283-329.
doi: doi:10.1016/j.anihpc.2005.03.003. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Intern. A, 1 (1937), 1-26. |
[20] |
J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20.
doi: doi:10.1137/S0036141093246105. |
[21] |
Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sin. (N.S.), 3 (2008), 567-584. |
[22] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Equations, 213 (2005), 204-233.
doi: doi:10.1016/j.jde.2004.06.011. |
[23] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Dis. Cont. Dyn. Systems, 15 (2006), 819-832.
doi: doi:10.3934/dcds.2006.15.819. |
[24] |
R. Pinsky, Second order elliptic operators with periodic coefficients: Criticality theory, perturbations, and positive harmonic functions, J. Func. Anal., 129 (1995), 80-107.
doi: doi:10.1006/jfan.1995.1043. |
[25] |
J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 14 (1997), 499-552.
doi: doi:10.1016/S0294-1449(97)80137-0. |
[26] |
J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233.
doi: doi:10.1007/s10231-008-0072-7. |
[27] |
A. Scheel, Coarsening fronts, Arch. Ration. Mech. Anal., 181 (2006), 505-534.
doi: doi:10.1007/s00205-006-0422-9. |
[28] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equation, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: doi:10.1137/060661788. |
[29] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Diff. Equations, 246 (2009), 2103-2130.
doi: doi:10.1016/j.jde.2008.06.037. |
[30] |
J. M. Vega, On the uniqueness of multidimensional travelling fronts of some semilinear equations, J. Math. Anal. Appl., 177 (1993), 481-490.
doi: doi:10.1006/jmaa.1993.1271. |
[31] |
J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Part. Diff. Equations, 18 (1993), 505-531.
doi: doi:10.1080/03605309308820939. |
[32] |
J. M. Vega, Multidimensional travelling fronts in a model from combustion theory and related problems, Diff. Int. Equations, 6 (1993), 131-155. |
[33] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: doi:10.1007/s00285-002-0169-3. |
show all references
References:
[1] |
H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. |
[2] |
H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Amer. Math. Soc., Contemp. Math., 446, (2007), 101-123. |
[3] |
H. Berestycki and F. Hamel, "Reaction-Diffusion Equations and Propagation Phenomena,", Applied Mathematical Sciences, ().
|
[4] |
H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: doi:10.1215/S0012-7094-00-10331-6. |
[5] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
doi: doi:10.1016/j.matpur.2004.10.006. |
[6] |
H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 9 (1992), 497-572. |
[7] |
G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Diff. Equations, 236 (2007), 237-279.
doi: doi:10.1016/j.jde.2007.01.021. |
[8] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 24 (2007), 369-393.
doi: doi:10.1016/j.anihpc.2006.03.012. |
[9] |
M. El Smaily, Min-max formulæ for the speeds of pulsating travelling fronts in periodic excitable media, Ann. Mat. Pura Appl., 189 (2010), 47-66.
doi: doi:10.1007/s10231-009-0100-2. |
[10] |
P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Philadelphia, PA, 1988. |
[11] |
P. C. Fife and J. B. McLeod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.
doi: doi:10.1007/BF00250432. |
[12] |
F. Hamel, Formules min-max pour les vitesses d'ondes progressives multidimensionnelles, Ann. Fac. Sci. Toulouse, 8 (1999), 259-280. |
[13] |
F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets, Comm. Part. Diff. Equations, 25 (2000), 769-819.
doi: doi:10.1080/03605300008821532. |
[14] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Scient. Ecole Norm. Sup., 37 (2004), 469-506. |
[15] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Disc. Cont. Dyn. Systems, 13 (2005), 1069-1096.
doi: doi:10.3934/dcds.2005.13.1069. |
[16] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Disc. Cont. Dyn. Systems, 14 (2006), 75-92. |
[17] |
F. Hamel and N. Nadirashvili, Travelling waves and entire solutions of the Fisher-KPP equation in $\R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: doi:10.1007/PL00004238. |
[18] |
M. Haragus, A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 23 (2006), 283-329.
doi: doi:10.1016/j.anihpc.2005.03.003. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Intern. A, 1 (1937), 1-26. |
[20] |
J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20.
doi: doi:10.1137/S0036141093246105. |
[21] |
Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sin. (N.S.), 3 (2008), 567-584. |
[22] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Equations, 213 (2005), 204-233.
doi: doi:10.1016/j.jde.2004.06.011. |
[23] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Dis. Cont. Dyn. Systems, 15 (2006), 819-832.
doi: doi:10.3934/dcds.2006.15.819. |
[24] |
R. Pinsky, Second order elliptic operators with periodic coefficients: Criticality theory, perturbations, and positive harmonic functions, J. Func. Anal., 129 (1995), 80-107.
doi: doi:10.1006/jfan.1995.1043. |
[25] |
J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 14 (1997), 499-552.
doi: doi:10.1016/S0294-1449(97)80137-0. |
[26] |
J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233.
doi: doi:10.1007/s10231-008-0072-7. |
[27] |
A. Scheel, Coarsening fronts, Arch. Ration. Mech. Anal., 181 (2006), 505-534.
doi: doi:10.1007/s00205-006-0422-9. |
[28] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equation, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: doi:10.1137/060661788. |
[29] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Diff. Equations, 246 (2009), 2103-2130.
doi: doi:10.1016/j.jde.2008.06.037. |
[30] |
J. M. Vega, On the uniqueness of multidimensional travelling fronts of some semilinear equations, J. Math. Anal. Appl., 177 (1993), 481-490.
doi: doi:10.1006/jmaa.1993.1271. |
[31] |
J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Part. Diff. Equations, 18 (1993), 505-531.
doi: doi:10.1080/03605309308820939. |
[32] |
J. M. Vega, Multidimensional travelling fronts in a model from combustion theory and related problems, Diff. Int. Equations, 6 (1993), 131-155. |
[33] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: doi:10.1007/s00285-002-0169-3. |
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