Absence of Self-Similar Blow-up and Local Well-posedness for the Constant Mean-Curvature Wave Equation

TL;DR

This paper proves that the constant-mean-curvature wave equation in 2D does not have self-similar blow-up solutions and establishes local well-posedness for certain initial data spaces.

Contribution

It demonstrates the absence of self-similar blow-up and confirms local well-posedness for initial data in the critical Sobolev space.

Findings

No self-similar blow-up solutions exist.

The equation is locally well-posed in b4H^{3/2}.

Provides insights into the equation's solution behavior.

Abstract

In this note, we consider the constant-mean-curvature wave equation in $(1+2)$-dimensions. We show that it does not admit any self-similar blow-up. We also remark that the equation is locally well-posed for initial data in $\dot{H}^{3/2}$.