# American Institute of Mathematical Sciences

May  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
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