Finite time blow-up for the heat flow of H-surface with constant mean curvature

TL;DR

This paper investigates conditions under which solutions to the heat flow of surfaces with constant mean curvature develop finite time singularities, extending previous results on global well-posedness and blow-up behavior.

Contribution

The paper provides new initial data conditions that guarantee finite time blow-up for solutions, complementing earlier work on global existence and singularity formation.

Findings

Identified new initial data conditions leading to finite time blow-up.

Extended previous results on heat flow of surfaces with constant mean curvature.

Demonstrated finite time singularity development under these conditions.

Abstract

In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures, Manuscripta Mathematica, 2011, 134: 259-271] by Huang et al., where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.