Abstract
This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values. We first show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infinity. Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.
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References
Aronson D G, Weinberger H F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 466. Berlin: Springer-Verlag, 1975, 5–49
Aronson D G, Weinberger H F. Multidimensional nonlinear diffusions arising in population genetics. Adv Math, 1978, 30: 33–76
Bonnet A, Hamel F. Existence of non-planar solutions of a simple model of premixed Bunsen flames. SIAM J Math Anal, 1999, 31: 80–118
Chen X, Guo J S, Hamel F, et al. Traveling waves with paraboloid like interfaces for balanced bistable dynamics. Ann Inst H Poincaré Anal Linéaire, 2007, 24: 369–393
Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd ed. New York: Springer-Verlag, 1983
Gui C. Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2. Arch Ration Mech Anal, 2011, 203: 1037–1065
Fife P C, McLeod J B. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch Ration Mech Anal, 1977, 65: 335–361
Fisher R A. The advance of advantageous genes. Ann Eugenics, 1937, 7: 355–369
Hamel F, Monneau R. Solutions of semilinear elliptic equations in ℝN with conical-shaped level sets. Comm Partial Differential Equations, 2000, 25: 769–819
Hamel F, Monneau R, Roquejoffre J M. Stability of travelling waves in a model for conical flames in two space dimensions. Ann Sci Ecole Norm Sup, 2004, 37: 469–506
Hamel F, Monneau R, Roquejoffre J M. Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin Dyn Syst, 2005, 13: 1069–1096
Hamel F, Monneau R, Roquejoffre J M. Asymptotic properties and classification of bistable fronts with Lipschitz level sets. Discrete Contin Dyn Syst, 2006, 14: 75–92
Hamel F, Nadirashvili N. Traveling fronts and entire solutions of the Fisher-KPP equation in ℝN. Arch Ration Mech Anal, 2001, 157: 91–163
Haragus M, Scheel A. Almost planar waves in anisotropic media. Comm Partial Differential Equations, 2006, 31: 791–815
Haragus M, Scheel A. Corner defects in almost planar interface propagation. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 283–329
Huang R. Stability of travelling fronts of the Fisher-KPP equation in ℝN. Nonlinear Differential Equations Appl, 2008, 15: 599–622
Kapitula T. Multidimensional stability of planar traveling waves. Trans Amer Math Soc, 1997, 349: 257–269
Kolmogorov A N, Petrovsky I G, Piskunov N S. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul Moskovskogo Gos Univ, 1937, 1: 1–26
Kurokawa Y, Taniguchi M. Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations. Proc Roy Soc Edinburgh Sect A, 2011, 141: 1–24
Levermore C D, Xin J. Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II. Comm Partial Differential Equations, 1992, 17: 1901–1924
Liu N W, Li W T. Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media. Sci China Math, 2010, 53: 1775–1786
Lü G Y, Wang M X. Stability of planar waves in mono-stable reaction-diffusion equations. Proc Amer Math Soc, 2011, 139: 3611–3621
Lü G Y, Wang M X. Stability of planar waves in reaction-diffusion system. Sci China Math, 2011, 54: 1403–1419
Matano H, Nara M. Large time behavior of disturbed planar fronts in the Allen-Cahn equation. J Differential Equations, 2011, 251: 3522–3557
Matano H, Nara M, Taniguchi M. Stability of planar waves in the Allen-Cahn equation. Comm Partial Differential Equations, 2009, 34: 976–1002
Nara M, Taniguchi M. Stability of a traveling wave in curvature flows for spatially non-decaying perturbations, Discrete Contin Dyn Syst, 2006, 14: 203–220
Nara M, Taniguchi M. Convergence to V-shaped fronts for spatially non-decaying inital perturbations. Discrete Contin Dyn Syst, 2006, 16: 137–156
Ninomiya H, Taniguchi M. Existence and global stability of traveling curved fronts in the Allen-Cahn equations. J Differential Equations, 2005, 213: 204–233
Ninomiya H, Taniguchi M. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete Contin Dyn Syst, 2006, 15: 819–832
Roquejoffre J M, Roussier-Michon V. Nontrivial large-time behaviour in bistable reaction-diffusion equations. Ann Mat Pura Appl, 2009, 188: 207–233
Sheng W J, Li W T, Wang Z C. Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity. J Differential Equations, 2012, 252: 2388–2424
Sheng W J, Li W T, Wang Z C, et al. Global exponential stability of V-shaped traveling fronts in the Allen-Cahn equation. J Dynam Differential Equations,2011, submitted
Shen W J, Wang M X. Global exponential stability of time-periodic V-shaped traveling fronts in bistable reactiondiffusion equations. Nonlinearity, 2012, submitted
El Smaily M, Hamel F, Huang R. Two-dimensional curved fronts in a periodic shear flow. Nonlinear Anal, 2011, 74: 6469–6486
Taniguchi M. Traveling fronts of pyramidal shapes in the Allen-Cahn equations. SIAM J Math Anal, 2007, 39: 319–344
Taniguchi M. The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations. J Differential Equations, 2009, 246: 2103–2130
Volpert A I, Volpert V A, Volpert V A. Travelling Wave Solutions of Parabolic Systems. In: Translations of Mathematical Monographs, vol. 140. Providence, RI: American Mathematical Society, 1994
Wang Z C, Wu J. Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity. J Differential Equations, 2011, 250: 3196–3229
Wang Z C. Traveling curved fronts in monotone bistable systems. Discrete Contin Dyn Syst, 2012, 32: 2339–2374
Wu Y P, Zhao Y. The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion. Sci China Math, 2010, 53: 1161–1184
Xin J. Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I. Comm Partial Differential Equations, 1992, 17: 1889–1899
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Sheng, W., Li, W. & Wang, Z. Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation. Sci. China Math. 56, 1969–1982 (2013). https://doi.org/10.1007/s11425-013-4699-5
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DOI: https://doi.org/10.1007/s11425-013-4699-5