On the entropy of parabolic Allen–Cahn equation

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