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 α∈(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/(αε)), 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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BibTeX RIS

BibTeX

@misc{cusseddu2026slow,
  title = {{Slow motion for the time-fractional Allen-Cahn equation with linear and nonlinear diffusion}},
  author = {Davide Cusseddu and Raffaele Folino and Luis Fernando López Ríos},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04208},
  eprint = {2610.04208},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04208},
}

RIS

TY  - UNPB
TI  - Slow motion for the time-fractional Allen-Cahn equation with linear and nonlinear diffusion
AU  - Davide Cusseddu
AU  - Raffaele Folino
AU  - Luis Fernando López Ríos
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04208
UR  - https://arxiv.org/abs/2610.04208
AB  - We study the slow motion of phase transition layers for a time-fractional Allen--Cahn equation with a Caputo time derivative of order α∈(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/(αε)), 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.
ER  -

Abstract

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