We define a local (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.

中文翻译：

---

关于抛物线 Allen-Cahn 方程的熵

我们为 $\mathbb{R}^n$ 或紧凑流形中的氡测量定义了一个局部（平均曲率流）熵。此外，我们证明了与抛物线 Allen-Cahn 方程相关的测度的熵的单调性公式。如果环境流形是具有非负截面曲率和平行 Ricci 曲率的紧流形，这是抛物线 Allen-Cahn 方程的新单调性公式的结果。作为一个应用，我们证明了当初始数据的熵足够小（小于一维驻波能量的两倍）时，抛物线 Allen-Cahn 方程的极限测度在所有未来时间都具有单位密度.