# American Institute of Mathematical Sciences

2016, 36(5): 2473-2496. doi: 10.3934/dcds.2016.36.2473

## Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

 1 CAMS - Ecole des Hautes Etudes en Sciences Sociales, 190-198 avenue de France, 75013 Paris, France

Received  November 2014 Revised  September 2015 Published  October 2015

We prove the multidimensional stability of planar traveling waves for scalar nonlocal Allen-Cahn equations using semigroup estimates. We show that if the traveling wave is spectrally stable in one space dimension, then it is stable in $n$-space dimension, $n\geq 2$, with perturbations of the traveling wave decaying like $t^{-(n-1)/4}$ as $t\rightarrow +\infty$ in $H^k(\mathbb{R}^n)$ for $k\geq \left[\frac{n+1}{2}\right]$.
Citation: Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473
##### References:
 [1] P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: 10.1016/S0022-247X(02)00205-6. [2] P. W. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equations,, SIAM J. Math. Anal., 38 (2006), 116. doi: 10.1137/S0036141004443968. [3] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. [4] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. [5] F. Chen, Uniform stability if multidimensional travelling waves for the nonlocal Allen-Cahn equation,, Electronic Journal of Differential Equations, 10 (2003), 109. [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Advences in Differential Equations, 2 (1997), 125. [7] J. Coville, Equation de Réaction Diffusion Non-locale,, Thèse de doctorat del'université Pierre et Marie Curie, (2003). [8] A. De Masi, T. Gobron and E. Presutti, Traveling fronts in non-local evolution equations,, Arch. Rat. Mech. Anal, 132 (1995), 143. doi: 10.1007/BF00380506. [9] G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network,, Proc. Roy. Soc. Edin., 123 (1993), 461. doi: 10.1017/S030821050002583X. [10] T. Gallay and A. Scheel, Diffusive stability of oscillations in reaction-diffusion systems,, Trans. Amer. Math. Soc., 363 (2011), 2571. doi: 10.1090/S0002-9947-2010-05148-7. [11] A. Hoffman, H. J. Hupkes and E. S. van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions,, Trans. Amer. Math. Soc., 367 (2015), 8757. doi: 10.1090/S0002-9947-2015-06392-2. [12] M. A. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case,, J. Differential Equations, 249 (2010), 1213. doi: 10.1016/j.jde.2010.04.015. [13] T. Kapitula, Multidimensional stability of planar traveling waves,, Trans. Amer. Math. Soc., 349 (1997), 257. doi: 10.1090/S0002-9947-97-01668-1. [14] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, II,, Comm. in Par. Diff. Equ., 17 (1992), 1901. doi: 10.1080/03605309208820908. [15] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. in Par. Diff. Equ., 34 (2009), 976. doi: 10.1080/03605300902963500. [16] J. R. Miller and H. Zeng, Multidimensional stability of planar traveling waves for an integrodifference model,, Discrete and Continuous Dynamical Systems B, 18 (2013), 741. [17] M. Oh and K. Zumbrun, Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions,, Arch. Ration. Mech. Anal., 196 (2010), 1. doi: 10.1007/s00205-009-0229-6. [18] J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I,, Comm. in Par. Diff. Equ., 17 (1992), 1889. doi: 10.1080/03605309208820907.

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##### References:
 [1] P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: 10.1016/S0022-247X(02)00205-6. [2] P. W. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equations,, SIAM J. Math. Anal., 38 (2006), 116. doi: 10.1137/S0036141004443968. [3] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. [4] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. [5] F. Chen, Uniform stability if multidimensional travelling waves for the nonlocal Allen-Cahn equation,, Electronic Journal of Differential Equations, 10 (2003), 109. [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Advences in Differential Equations, 2 (1997), 125. [7] J. Coville, Equation de Réaction Diffusion Non-locale,, Thèse de doctorat del'université Pierre et Marie Curie, (2003). [8] A. De Masi, T. Gobron and E. Presutti, Traveling fronts in non-local evolution equations,, Arch. Rat. Mech. Anal, 132 (1995), 143. doi: 10.1007/BF00380506. [9] G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network,, Proc. Roy. Soc. Edin., 123 (1993), 461. doi: 10.1017/S030821050002583X. [10] T. Gallay and A. Scheel, Diffusive stability of oscillations in reaction-diffusion systems,, Trans. Amer. Math. Soc., 363 (2011), 2571. doi: 10.1090/S0002-9947-2010-05148-7. [11] A. Hoffman, H. J. Hupkes and E. S. van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions,, Trans. Amer. Math. Soc., 367 (2015), 8757. doi: 10.1090/S0002-9947-2015-06392-2. [12] M. A. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case,, J. Differential Equations, 249 (2010), 1213. doi: 10.1016/j.jde.2010.04.015. [13] T. Kapitula, Multidimensional stability of planar traveling waves,, Trans. Amer. Math. Soc., 349 (1997), 257. doi: 10.1090/S0002-9947-97-01668-1. [14] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, II,, Comm. in Par. Diff. Equ., 17 (1992), 1901. doi: 10.1080/03605309208820908. [15] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation,, Comm. in Par. Diff. Equ., 34 (2009), 976. doi: 10.1080/03605300902963500. [16] J. R. Miller and H. Zeng, Multidimensional stability of planar traveling waves for an integrodifference model,, Discrete and Continuous Dynamical Systems B, 18 (2013), 741. [17] M. Oh and K. Zumbrun, Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions,, Arch. Ration. Mech. Anal., 196 (2010), 1. doi: 10.1007/s00205-009-0229-6. [18] J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I,, Comm. in Par. Diff. Equ., 17 (1992), 1889. doi: 10.1080/03605309208820907.
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