Metastable Speeds in the Fractional AllenCahn Equation
Abstract
We study numerically the onedimensional AllenCahn equation with the spectral fractional Laplacian $(\Delta)^{\alpha/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $\alpha$ of the fractional Laplacian. For the associated model on the realline we derive asymptotic formulas for the interface speed and timetocollision in terms of $\alpha$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finiteelement method based upon extending the fractional Laplacian to a cylinder in the upperhalf plane, and compute the interface speed, timetocollapse and interface width for $\alpha\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and timetocollision give a good approximation for large intervals.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.02731
 Bibcode:
 2020arXiv200602731A
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Numerical Analysis