Wave equations associated to Liouville systems and constant mean curvature equations

TL;DR

This paper investigates wave equations related to Liouville systems and constant mean curvature equations in two dimensions, establishing blow-up criteria and finite-time blow-up results using conformal geometry techniques.

Contribution

It introduces new blow-up criteria for the wave Liouville equation and demonstrates finite-time blow-up for constant mean curvature wave equations under specific initial conditions.

Findings

Blow-up criteria for wave Liouville equations established.

Finite-time blow-up demonstrated for constant mean curvature equations.

Tools from conformal geometry are used to analyze blow-up behavior.

Abstract

We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial data.