Interacting fronts in the strongly nonlocal Allen--Cahn equation

Abstract: We study the sharp-interface limit of the fractional Allen--Cahn equation in $\R<sup>n$, $n\geq 2$, in the strongly nonlocal regime $s\in(0,\frac12)$, for initial data consisting of finitely many nested transition layers. We identify a coupled geometric law for the motion of the resulting fronts: each velocity contains fractional mean curvature and a nonlocal interaction potential generated by the other fronts. These interactions persist while the fronts remain separated, in contrast to the independent motion in the critical regime $s=\frac12$. Assuming that the coupled law admits a smooth evolution with strictly nested sets on a given time interval, we prove that the Allen--Cahn solutions converge locally uniformly away from the moving fronts to the corresponding integer-valued phases.