发表机构
Dipartimento di Scienze Matematiche “G. L. Lagrange” Politecnico di Torino; Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México（都灵理工大学数学科学系; 墨西哥国立自治大学应用数学与系统研究所数学与力学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究时间分数阶 Allen-Cahn 方程的慢运动，证明分数阶记忆区分非退化与退化势的指数/代数慢动力学，并推广至非线性扩散算子。
AI 中文摘要
我们研究了具有 Caputo 时间导数阶数 $\alpha\in(0,1)$ 的时间分数阶 Allen-Cahn 方程中相变层的慢运动。我们证明分数阶记忆保持了与非退化势和退化势分别相关的指数慢动力学和代数慢动力学之间的区别。在非退化情形下，分层解在阶为 $\exp(A/(\alpha\varepsilon))$（$A>0$）的指数长时间尺度上持续存在，而在退化情形下，相应的代数时间尺度被因子 $1/\alpha$ 修改。最后，我们将分析扩展到非线性扩散算子，考虑 $p$-Laplacian 和 Perona-Malik 型扩散，证明相同的分数阶慢运动机制在这些更一般的设置中仍然存在。
英文摘要
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.
Comments26 pages, 2 figures
