Formation of singularities for the relativistic membrane equation with radial symmetry

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].