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Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation
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 |
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, In Goldstein J, 466 (1975), 5.
|
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math, 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.
|
[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.
doi: 10.1016/j.anihpc.2006.03.012. |
[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.
|
[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.
doi: 10.1016/j.ansens.2004.03.001. |
[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.
doi: 10.3934/dcds.2005.13.1069. |
[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.
|
[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.
doi: 10.1017/S0308210510001253. |
[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.
doi: 10.1007/s11425-010-4032-5. |
[11] |
H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation,, J. Differential Equations, 251 (2011), 3522.
doi: 10.1016/j.jde.2011.08.029. |
[12] |
H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. Partial Differential Equations, 34 (2009), 976.
doi: 10.1080/03605300902963500. |
[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.
doi: 10.1016/j.jde.2004.06.011. |
[14] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, Discrete Contin. Dyn. Syst., 15 (2006), 819.
doi: 10.3934/dcds.2006.15.819. |
[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.
|
[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.
doi: 10.3934/dcds.2006.16.137. |
[17] |
W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion equations,, Netw. Heterog. Media, 8 (2013), 379.
doi: 10.3934/nhm.2013.8.379. |
[18] |
J. M. Roquejoffre and V. M. Roussier, Nontrivial large-time behaviour in bistable reaction-diffusion equations,, Ann. Mat. Pura Appl., 188 (2009), 207.
doi: 10.1007/s10231-008-0072-7. |
[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.
doi: 10.1016/j.jde.2011.09.016. |
[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.
doi: 10.1007/s11425-013-4699-5. |
[21] |
D. Terman, Directed graphs and traveling waves,, Trans. Amer. Math. Soc., 289 (1985), 809.
doi: 10.1090/S0002-9947-1985-0784015-6. |
[22] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319.
doi: 10.1137/060661788. |
[23] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, J. Differential Equations, 246 (2009), 2103.
doi: 10.1016/j.jde.2008.06.037. |
[24] |
M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst., 32 (2012), 1011.
doi: 10.3934/dcds.2012.32.1011. |
[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.
doi: 10.1016/j.jde.2011.01.017. |
[26] |
Z.-C. Wang., Traveling curved fronts in monotone bistable systems,, Distere Contin. Dyn. Syst., 32 (2012), 2339.
doi: 10.3934/dcds.2012.32.2339. |
[27] |
J.-X. Xin., Multidimensional stability of traveling waves in a bistable reaction-diffusion equation,, I. Comm. Partial Differential Equations, 17 (1992), 1889.
doi: 10.1080/03605309208820907. |
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, 466 (1975), 5.
|
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math, 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.
|
[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.
doi: 10.1016/j.anihpc.2006.03.012. |
[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.
|
[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.
doi: 10.1016/j.ansens.2004.03.001. |
[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.
doi: 10.3934/dcds.2005.13.1069. |
[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.
|
[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.
doi: 10.1017/S0308210510001253. |
[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.
doi: 10.1007/s11425-010-4032-5. |
[11] |
H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation,, J. Differential Equations, 251 (2011), 3522.
doi: 10.1016/j.jde.2011.08.029. |
[12] |
H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. Partial Differential Equations, 34 (2009), 976.
doi: 10.1080/03605300902963500. |
[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.
doi: 10.1016/j.jde.2004.06.011. |
[14] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, Discrete Contin. Dyn. Syst., 15 (2006), 819.
doi: 10.3934/dcds.2006.15.819. |
[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.
|
[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.
doi: 10.3934/dcds.2006.16.137. |
[17] |
W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion equations,, Netw. Heterog. Media, 8 (2013), 379.
doi: 10.3934/nhm.2013.8.379. |
[18] |
J. M. Roquejoffre and V. M. Roussier, Nontrivial large-time behaviour in bistable reaction-diffusion equations,, Ann. Mat. Pura Appl., 188 (2009), 207.
doi: 10.1007/s10231-008-0072-7. |
[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.
doi: 10.1016/j.jde.2011.09.016. |
[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.
doi: 10.1007/s11425-013-4699-5. |
[21] |
D. Terman, Directed graphs and traveling waves,, Trans. Amer. Math. Soc., 289 (1985), 809.
doi: 10.1090/S0002-9947-1985-0784015-6. |
[22] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, SIAM J. Math. Anal., 39 (2007), 319.
doi: 10.1137/060661788. |
[23] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, J. Differential Equations, 246 (2009), 2103.
doi: 10.1016/j.jde.2008.06.037. |
[24] |
M. Taniguchi, Multi-Dimensional traveling fronts in bistable reaction-diffusion equations,, Discrete Contin. Dyn. Syst., 32 (2012), 1011.
doi: 10.3934/dcds.2012.32.1011. |
[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.
doi: 10.1016/j.jde.2011.01.017. |
[26] |
Z.-C. Wang., Traveling curved fronts in monotone bistable systems,, Distere Contin. Dyn. Syst., 32 (2012), 2339.
doi: 10.3934/dcds.2012.32.2339. |
[27] |
J.-X. Xin., Multidimensional stability of traveling waves in a bistable reaction-diffusion equation,, I. Comm. Partial Differential Equations, 17 (1992), 1889.
doi: 10.1080/03605309208820907. |
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