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arXiv · 2610.04208

Slow motion for the time-fractional Allen-Cahn equation with linear and nonlinear diffusion

Davide Cusseddu·Raffaele Folino·Luis Fernando López Ríos
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Abstract

We study the slow motion of phase transition layers for a time-fractional Allen--Cahn equation with a Caputo time derivative of order $α\in(0,1)$. We show that the fractional memory preserves the distinction between exponentially and algebraically slow dynamics associated with non-degenerate and degenerate potentials, respectively. In the non-degenerate case, layered solutions persist on an exponentially long time scale of order $\exp(A/(α\varepsilon))$, $A>0$, while in the degenerate case the corresponding algebraic time scale is modified by the factor $1/α$. Finally, we extend the analysis to nonlinear diffusion operators, considering both the $p$-Laplacian and a Perona--Malik type diffusion, showing that the same fractional slow-motion mechanism persists in these more general settings.

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Davide Cusseddu, Raffaele Folino, Luis Fernando López Ríos. 2026-10-03. Slow motion for the time-fractional Allen-Cahn equation with linear and nonlinear diffusion. https://arxiv.org/abs/2610.04208

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