  Cahn-Hilliard models are central for describing the evolution of interfaces
in phase separation processes and free boundary problems. In general, they have
non-constant and often degenerate mobilities. However, in the latter case, the
spontaneous appearance of points of vanishing mobility and their impact on the
solution are not well understood. In this paper we develop a singular
perturbation theory to identify a range of degeneracies for which the solution
of the Cahn-Hilliard equation forms a singularity in infinite time. This
analysis forms the basis for a rigorous sharp interface theory and enables the
systematic development of robust numerical methods for this family of model
equations.
