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

Abstract

We study the slow motion of phase transition layers for a time-fractional Allen--Cahn equation with a Caputo time derivative of order $\alpha\in(0,1)$. 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 $\exp(A/(\alpha\varepsilon))$, $A>0$, while in the degenerate case the corresponding algebraic time scale is modified by the factor $1/\alpha$. 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.