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.
Reference
J. Differential Equations 265 (2018), no. 4, 1528–1575