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Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation

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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

  1. 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

    Chapter  Google Scholar 

  2. Aronson D G, Weinberger H F. Multidimensional nonlinear diffusions arising in population genetics. Adv Math, 1978, 30: 33–76

    Article  MathSciNet  MATH  Google Scholar 

  3. 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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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

    Article  MathSciNet  MATH  Google Scholar 

  5. Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd ed. New York: Springer-Verlag, 1983

    Book  MATH  Google Scholar 

  6. Gui C. Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2. Arch Ration Mech Anal, 2011, 203: 1037–1065

    Article  MathSciNet  Google Scholar 

  7. 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

    Article  MathSciNet  MATH  Google Scholar 

  8. Fisher R A. The advance of advantageous genes. Ann Eugenics, 1937, 7: 355–369

    Article  Google Scholar 

  9. Hamel F, Monneau R. Solutions of semilinear elliptic equations in ℝN with conical-shaped level sets. Comm Partial Differential Equations, 2000, 25: 769–819

    Article  MathSciNet  MATH  Google Scholar 

  10. 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

    MathSciNet  MATH  Google Scholar 

  11. Hamel F, Monneau R, Roquejoffre J M. Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin Dyn Syst, 2005, 13: 1069–1096

    Article  MathSciNet  MATH  Google Scholar 

  12. 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

    MathSciNet  MATH  Google Scholar 

  13. Hamel F, Nadirashvili N. Traveling fronts and entire solutions of the Fisher-KPP equation in ℝN. Arch Ration Mech Anal, 2001, 157: 91–163

    Article  MathSciNet  MATH  Google Scholar 

  14. Haragus M, Scheel A. Almost planar waves in anisotropic media. Comm Partial Differential Equations, 2006, 31: 791–815

    Article  MathSciNet  MATH  Google Scholar 

  15. Haragus M, Scheel A. Corner defects in almost planar interface propagation. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 283–329

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang R. Stability of travelling fronts of the Fisher-KPP equation in ℝN. Nonlinear Differential Equations Appl, 2008, 15: 599–622

    Article  MATH  Google Scholar 

  17. Kapitula T. Multidimensional stability of planar traveling waves. Trans Amer Math Soc, 1997, 349: 257–269

    Article  MathSciNet  MATH  Google Scholar 

  18. 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

    Google Scholar 

  19. Kurokawa Y, Taniguchi M. Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations. Proc Roy Soc Edinburgh Sect A, 2011, 141: 1–24

    Article  MathSciNet  Google Scholar 

  20. Levermore C D, **n J. Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II. Comm Partial Differential Equations, 1992, 17: 1901–1924

    Article  MathSciNet  MATH  Google Scholar 

  21. 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

    Article  MathSciNet  MATH  Google Scholar 

  22. Lü G Y, Wang M X. Stability of planar waves in mono-stable reaction-diffusion equations. Proc Amer Math Soc, 2011, 139: 3611–3621

    Article  MathSciNet  Google Scholar 

  23. Lü G Y, Wang M X. Stability of planar waves in reaction-diffusion system. Sci China Math, 2011, 54: 1403–1419

    Article  MathSciNet  MATH  Google Scholar 

  24. Matano H, Nara M. Large time behavior of disturbed planar fronts in the Allen-Cahn equation. J Differential Equations, 2011, 251: 3522–3557

    Article  MathSciNet  MATH  Google Scholar 

  25. Matano H, Nara M, Taniguchi M. Stability of planar waves in the Allen-Cahn equation. Comm Partial Differential Equations, 2009, 34: 976–1002

    Article  MathSciNet  MATH  Google Scholar 

  26. 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

    MathSciNet  MATH  Google Scholar 

  27. Nara M, Taniguchi M. Convergence to V-shaped fronts for spatially non-decaying inital perturbations. Discrete Contin Dyn Syst, 2006, 16: 137–156

    Article  MathSciNet  MATH  Google Scholar 

  28. Ninomiya H, Taniguchi M. Existence and global stability of traveling curved fronts in the Allen-Cahn equations. J Differential Equations, 2005, 213: 204–233

    Article  MathSciNet  MATH  Google Scholar 

  29. Ninomiya H, Taniguchi M. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete Contin Dyn Syst, 2006, 15: 819–832

    Article  MathSciNet  MATH  Google Scholar 

  30. Roquejoffre J M, Roussier-Michon V. Nontrivial large-time behaviour in bistable reaction-diffusion equations. Ann Mat Pura Appl, 2009, 188: 207–233

    Article  MathSciNet  MATH  Google Scholar 

  31. 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

    Article  MathSciNet  MATH  Google Scholar 

  32. 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

    Google Scholar 

  33. Shen W J, Wang M X. Global exponential stability of time-periodic V-shaped traveling fronts in bistable reactiondiffusion equations. Nonlinearity, 2012, submitted

    Google Scholar 

  34. El Smaily M, Hamel F, Huang R. Two-dimensional curved fronts in a periodic shear flow. Nonlinear Anal, 2011, 74: 6469–6486

    Article  MathSciNet  MATH  Google Scholar 

  35. Taniguchi M. Traveling fronts of pyramidal shapes in the Allen-Cahn equations. SIAM J Math Anal, 2007, 39: 319–344

    Article  MathSciNet  MATH  Google Scholar 

  36. Taniguchi M. The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations. J Differential Equations, 2009, 246: 2103–2130

    Article  MathSciNet  MATH  Google Scholar 

  37. 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

    Google Scholar 

  38. 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

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang Z C. Traveling curved fronts in monotone bistable systems. Discrete Contin Dyn Syst, 2012, 32: 2339–2374

    Article  MathSciNet  MATH  Google Scholar 

  40. 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

    Article  MathSciNet  MATH  Google Scholar 

  41. **n J. Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I. Comm Partial Differential Equations, 1992, 17: 1889–1899

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Natural Science Research Center, Harbin Institute of Technology, Harbin, 150080, China

    WeiJie Sheng

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    WeiJie Sheng, WanTong Li & ZhiCheng Wang

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  1. WeiJie Sheng
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  2. WanTong Li
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  3. ZhiCheng Wang
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Correspondence to WanTong Li.

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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

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