  We consider the Cahn-Hilliard equation on a manifold with conical
singularities and show the existence of bounded imaginary powers for suitable
closed extensions of the bilaplacian. Combining results and methods from
singular analysis with a theorem of Clement and Li we then prove the short time
solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and
obtain the asymptotics of the solution near the conical points. We deduce, in
particular, that regularity is preserved on the smooth part of the manifold and
singularities remain confined to the conical points. We finally show how the
Allen-Cahn equation can be treated by simpler considerations. Again we obtain
short time solvability and the behavior near the conical points.
