Metastable speeds in the fractional Allen–Cahn equation

We study numerically the one-dimensional Allen–Cahn equation with spectral fractional Laplacian (−Δ)α/2 on intervals homogeneous Neumann boundary conditions. In particular, we are interested in speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow case classical Laplacian. Here investigate how width change if vary exponent α For associated model real line derive asymptotic formulas for interface time–to–collision terms a scaling parameter ε. use numerical approach via finite-element method based upon extending cylinder upper-half plane, compute speed, time–to–collapse α∈(0.3,2]. A comparison shows that give good approximation large intervals.