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 theWillmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. This linear analysis explains the behaviour 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. Beside that we 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.

@Article{BCMS08,
  author       = {Burger, M. and Chu, S.-Y. and Markowich, P. and Schönlieb, C.},
  title        = {The Willmore functional and instabilities in the Cahn-Hilliard equation},
  journal      = {Commun. Math. Sci.},
  volume       = {6},
  pages        = {309-329},
  year         = {2008},
  url          = \{/2008/BCMS08},
}