Metastability of the Cahn-Hilliard equation in one space dimension

We establish metastability of the one-dimensional Cahn-Hilliard equation for initial data that is order-one in energy and order-one in 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 Λ, there are three phases of evolution: (1) the solution is drawn after a time of order Λ2 into an algebraically small neighborhood of the N-layer branch of the slow manifold, (2) the solution is drawn after a time of order Λ3 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.