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February  2011, 4(1): 101-123. doi: 10.3934/dcdss.2011.4.101

Heteroclinic connections for multidimensional bistable reaction-diffusion equations

François Hamel 1, and  Jean-Michel Roquejoffre 2,

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

Received  November 2009 Revised  January 2010 Published  October 2010

In this paper, non-planar two-dimensional travelling fronts connecting an unstable one-dimensional periodic limiting state to a constant stable state are constructed for some reaction-diffusion equations with bistable nonlinearities. The minimal speeds are characterized in terms of the spatial period of the unstable limiting state. The limits of the minimal speeds and of the travelling fronts as the period converges to a critical minimal value or to infinity are analyzed. The fronts converge to flat fronts or to some curved fronts connecting an unstable ground state to a constant stable state.
Keywords: curved fronts., bistable nonlinearity, Travelling fronts, reaction-diffusion equation, monostable connections.
Mathematics Subject Classification: 35B10, 35C07, 35J6.
Citation: François Hamel, Jean-Michel Roquejoffre. Heteroclinic connections for multidimensional bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 101-123. doi: 10.3934/dcdss.2011.4.101
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, Springer-Verlag, to appear.

[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 $\mathbb{R}^{N}$, 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, Springer-Verlag, to appear.

[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 $\mathbb{R}^{N}$, 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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