Abstract
In this paper we are interested in the finite-time stability of transition solutions of the
Cahn-Hilliard equation and its connection to the Willmore functional. We show that the Willmore
functional locally decreases or increases in time in the linearly stable or unstable case respectively.
This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation.
We perform numerical examples in one and two dimensions and show that in the neighbourhood
of transition solutions local instabilities occur in finite time. We also show convergence of solutions
of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic
decay of the Willmore functional in time.