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

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  • 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 $k\geq \left[\frac{n+1}{2}\right]$.
    Mathematics Subject Classification: Primary: 35K57, 34K20 and 47D06.

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