# Metastability of the Cahn-Hilliard equation in one space dimension

## TL;DR

This paper proves the metastability behavior of solutions to the one-dimensional Cahn-Hilliard equation, showing how solutions evolve through three phases and remain close to a slow manifold for exponentially long times.

## Contribution

It rigorously characterizes the metastable dynamics of the 1D Cahn-Hilliard equation, including precise timescales and neighborhoods near the slow manifold.

## Key findings

- Solutions reach an algebraically small neighborhood of the slow manifold in order erence
- Solutions enter an exponentially small neighborhood after order erence
- Solutions stay exponentially close for exponentially long times

## Abstract

We establish metastability of the one-dimensional Cahn-Hilliard equation for initial data that is order-one in energy and order-one in $\dot{H}^{-1}$ away from a point on the so-called slow manifold with $N$ well-separated layers. Specifically, we show that, for such initial data on a system of lengthscale $\Lambda$, there are three phases of evolution: (1) the solution is drawn after a time of order $\Lambda^2$ into an algebraically small neighborhood of the $N$-layer branch of the slow manifold, (2) the solution is drawn after a time of order $\Lambda^3$ into an exponentially small neighborhood of the $N$-layer branch of the slow manifold, (3) the solution is trapped for an exponentially long time exponentially close to the $N$-layer branch of the slow manifold. The timescale in phase (3) is obtained with the sharp constant in the exponential.