Smoothness and long time existence for solutions of the Cahn–Hilliard equation on manifolds with conical singularities

Abstract

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.