Metastability of almost minimizers in a 1-d Allen-Cahn equation on a compact domain

We study metastability of the one-dimensional Allen-Cahn equation on a bounded interval with pinned f1 boundary conditions. We are particularly interested in considering the case of an asymmetric, nondegenerate double-well potential, meaning that the second derivative of the potential is nonzero in each of the wells but G ''(-1) not equal G ''(1) is allowed. Under fairly general conditions on the initial data, we show that the solution is drawn quickly into a small neighborhood of the socalled slow manifold and remains stuck there for an exponentially long time (which we quantify), even though the unique global minimizer of the energy is far away. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).