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

## TL;DR

This paper proves that solutions to the Cahn-Hilliard equation on manifolds with conical singularities exist globally, become instantly smooth, and their behavior near singularities can be precisely characterized.

## Contribution

It establishes maximal L^q-regularity and instant smoothing for the Cahn-Hilliard equation on conical manifolds, with detailed asymptotic analysis near singularities.

## Key findings

- Solutions exist globally in time.
- Solutions become instantly smooth in space and time.
- Asymptotic behavior near conical tips is characterized.

## 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.