The Swift-Hohenberg equation on conic manifolds

TL;DR

This paper studies the Swift-Hohenberg equation on manifolds with conical singularities, establishing existence, uniqueness, regularity, and long-term behavior of solutions, and relating solutions' asymptotics to local geometry.

Contribution

It develops a framework for analyzing the Swift-Hohenberg equation on conic manifolds, including conditions for global existence and detailed asymptotic analysis near singularities.

Findings

Existence and uniqueness of short-time solutions in Mellin-Sobolev spaces.

Characterization of conditions for solutions to exist for all time.

Asymptotic expansion of solutions near singularities related to local geometry.

Abstract

We consider the Swift-Hohenberg equation on manifolds with conical singularities and show existence, uniqueness and maximal regularity of the short time solution in terms of Mellin-Sobolev spaces. Moreover, we give a necessary and sufficient condition so that the above solution exists for all times. Space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown. The same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory.