  We consider the Cahn-Hilliard equation on manifolds with conical
singularities. For appropriate initial data, we show that the solution exists
in the maximal $L^q$-regularity space for all times and becomes instantaneously
smooth in space and time, where the maximal $L^q$-regularity is obtained in the
sense of Mellin-Sobolev spaces. Moreover, we provide precise information
concerning the asymptotic behavior of the solution close to the conical tips in
terms of the local geometry.
