On determinism and well-posedness in multiple time dimensions

Abstract

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial hypersurfaces surface of arbitrary space-time signature. We show that, under a non-local constraint, the initial value problem posed on codimension-one hypersurfaces-the Cauchy problem-has global unique solutions in the Sobolev spaces Hm. Thus, it is well-posed. However, we show that the initial value problem on higher codimension hypersurfaces is ill-posed due to failure of uniqueness, at least when specifying a finite number of derivatives of the data. This failure is in contrast to a uniqueness result for data given in an arbitrary neighbourhood of such initial hypersurfaces, which Courant deduces from Asgeirsson's mean value theorem. We give a generalization of Courant's theorem that extends to a broader class of equations. The proofs use Fourier synthesis and the Holmgren-John uniqueness theorem. This journal is © 2009 The Royal Society.