Formation of singularities for the relativistic membrane equation with radial symmetry

TL;DR

This paper proves that singularities form in the relativistic membrane equation with radial symmetry when the hypersurface transitions from timelike to null, extending previous one-dimensional results.

Contribution

It introduces a new blow-up theorem using characteristic decomposition, demonstrating singularity formation in the relativistic membrane equation with radial symmetry.

Findings

Singularities occur when the hypersurface becomes null

The blow-up theorem applies to the relativistic membrane equation

Generalizes previous one-dimensional singularity results

Abstract

The relativistic membrane equation can be rewritten as a first order hyperbolic system. Making use of the characteristic decomposition method, a new blow-up theorem is established. As an application, it demonstrates the formation of singularities for the relativistic membrane equation. Indeed, the singularity occurs when the hypersurface turns from being timelike to being null. This generalizes the result of Kong, Sun and Zhou's work for one-dimensional case [J Math Phys 47(1): 013503, 2006].