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

Grégory  Faye

doi:10.3934/dcds.2016.36.2473

Article Contents

2016, Volume 36, Issue 5: 2473-2496. Doi: 10.3934/dcds.2016.36.2473

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

Grégory Faye^1,

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

Received: October 31, 2014

Revised: August 31, 2015

Published: May 2017

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Abstract

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 $k\geq \left[\frac{n+1}{2}\right]$.

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Mathematics Subject Classification: Primary: 35K57, 34K20 and 47D06.

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