# American Institute of Mathematical Sciences

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

 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.
Citation: Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and 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. [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. [3] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] D. Terman, Directed graphs and traveling waves, Trans. Amer. Math. Soc., 289 (1985), 809-847. 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), 319-344. 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), 2103-2130. 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), 1011-1046. 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), 3196-3229. 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), 2339-2374. 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), 1889-1899. doi: 10.1080/03605309208820907.

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##### 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. [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. [3] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] D. Terman, Directed graphs and traveling waves, Trans. Amer. Math. Soc., 289 (1985), 809-847. 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), 319-344. 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), 2103-2130. 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), 1011-1046. 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), 3196-3229. 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), 2339-2374. 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), 1889-1899. doi: 10.1080/03605309208820907.
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