On the entropy of parabolic Allen-Cahn equation (Q2244865)

Summary: 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.