Global existence and finite time blow-up for the heat flow of H-system with constant mean curvature

TL;DR

This paper investigates the long-term behavior of solutions to the heat flow of H-systems with constant mean curvature, establishing conditions for global existence or finite time blow-up based on initial energy levels.

Contribution

It extends recent results by providing a comprehensive analysis of global existence and blow-up conditions for the heat flow of H-systems using the modified potential well method.

Findings

Global existence for low or critical initial energy with decay rates.

Finite time blow-up for high initial energy under certain conditions.

Threshold results distinguishing between global solutions and blow-up scenarios.

Abstract

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [12].