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F. Achleitner, C. Kuehn, J. Melenk, A. Rieder:
"Metastable Speeds in the Fractional Allen-Cahn Equation";
in: "ASC Report 13/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 24.

We study numerically the one-dimensional Allen-Cahn equation with the spectral frac-tional Laplacian (−∆)α/2 on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilat- ing 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 α of the
fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for
the interface speed and time-to-collision in terms of α and a scaling parameter ε. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian
to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for α ∈ (0.2, 2]. A com- parison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

http://www.asc.tuwien.ac.at/preprint/2020/asc13x2020.pdf