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June  2015, 20(4): 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation

Hongmei Cheng 1, and  Rong Yuan 2,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received  April 2014 Revised  October 2014 Published  February 2015

This paper is concerned with the asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equation on $\mathbb{R}^n$, $n\geq 4$. Our first result states that pyramidal traveling fronts are asymptotically stable under the initial perturbations that decay at space infinity. Then we further show the existence of a solution that oscillates permanently between two pyramidal traveling fronts, which implies that pyramidal traveling fronts are not asymptotically stable under more general perturbations. Our main technique is the supersolution and subsolution method coupled with the comparison principle.
Keywords: multidimensional stability, asymptotic stability, Allen-Cahn equation, the comparison principle., pyramidal traveling fronts.
Mathematics Subject Classification: 35K57, 35B10, 35B35, 35C0.
Citation: Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math, 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[4]

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 Anal. Non Linéaire, 24 (2007), 369-393. doi: 10.1016/j.anihpc.2006.03.012.  Google Scholar

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P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  Google Scholar

[6]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001.  Google Scholar

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F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.  Google Scholar

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F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.  Google Scholar

[9]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proceedings of the Royal Society of Edinburgh, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.  Google Scholar

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H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557. doi: 10.1016/j.jde.2011.08.029.  Google Scholar

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H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500.  Google Scholar

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H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.  Google Scholar

[14]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.  Google Scholar

[15]

M. Nara and M. Taniguchi, Stability of a traveling wave in curvature flows for spatially non-decaying perturbations, Discrete Contin. Dyn. Syst., 14 (2006), 203-220.  Google Scholar

[16]

M. Nara and M. Taniguchi, Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations, Discrete Contin. Dyn. Syst., 16 (2006), 137-156. doi: 10.3934/dcds.2006.16.137.  Google Scholar

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W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion equations, Netw. Heterog. Media, 8 (2013), 379-395. doi: 10.3934/nhm.2013.8.379.  Google Scholar

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J. M. Roquejoffre and V. M. Roussier, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233. doi: 10.1007/s10231-008-0072-7.  Google Scholar

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W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.016.  Google Scholar

[20]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5.  Google Scholar

[21]

D. Terman, Directed graphs and traveling waves, Trans. Amer. Math. Soc., 289 (1985), 809-847. doi: 10.1090/S0002-9947-1985-0784015-6.  Google Scholar

[22]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.  Google Scholar

[23]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[24]

M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011.  Google Scholar

[25]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017.  Google Scholar

[26]

Z.-C. Wang., Traveling curved fronts in monotone bistable systems, Distere Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.  Google Scholar

[27]

J.-X. Xin., Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I. Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math, 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[4]

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 Anal. Non Linéaire, 24 (2007), 369-393. doi: 10.1016/j.anihpc.2006.03.012.  Google Scholar

[5]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  Google Scholar

[6]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001.  Google Scholar

[7]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.  Google Scholar

[8]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.  Google Scholar

[9]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proceedings of the Royal Society of Edinburgh, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.  Google Scholar

[10]

N.-W. Liu and W.-T. Li, Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media, Sci. China Math., 53 (2010), 1775-1786. doi: 10.1007/s11425-010-4032-5.  Google Scholar

[11]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557. doi: 10.1016/j.jde.2011.08.029.  Google Scholar

[12]

H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500.  Google Scholar

[13]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.  Google Scholar

[14]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.  Google Scholar

[15]

M. Nara and M. Taniguchi, Stability of a traveling wave in curvature flows for spatially non-decaying perturbations, Discrete Contin. Dyn. Syst., 14 (2006), 203-220.  Google Scholar

[16]

M. Nara and M. Taniguchi, Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations, Discrete Contin. Dyn. Syst., 16 (2006), 137-156. doi: 10.3934/dcds.2006.16.137.  Google Scholar

[17]

W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion equations, Netw. Heterog. Media, 8 (2013), 379-395. doi: 10.3934/nhm.2013.8.379.  Google Scholar

[18]

J. M. Roquejoffre and V. M. Roussier, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233. doi: 10.1007/s10231-008-0072-7.  Google Scholar

[19]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.016.  Google Scholar

[20]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5.  Google Scholar

[21]

D. Terman, Directed graphs and traveling waves, Trans. Amer. Math. Soc., 289 (1985), 809-847. doi: 10.1090/S0002-9947-1985-0784015-6.  Google Scholar

[22]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.  Google Scholar

[23]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[24]

M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011.  Google Scholar

[25]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017.  Google Scholar

[26]

Z.-C. Wang., Traveling curved fronts in monotone bistable systems, Distere Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.  Google Scholar

[27]

J.-X. Xin., Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I. Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907.  Google Scholar

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