On determinism and well-posedness in multiple time dimensions

TL;DR

This paper investigates the well-posedness of initial value problems for wave and ultrahyperbolic equations in multiple time dimensions, establishing conditions for existence and uniqueness of solutions on various hypersurfaces.

Contribution

It demonstrates well-posedness under a nonlocal constraint on codimension-one hypersurfaces and highlights ill-posedness on higher codimension hypersurfaces, contrasting with classical results.

Findings

Well-posedness on codimension-one hypersurfaces with nonlocal constraints

Ill-posedness on higher codimension hypersurfaces for finite derivative data

Independent derivation of Courant and Hilbert's uniqueness result

Abstract

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on codimension-one hypersurfaces has global unique solutions in the Sobolev spaces $H^{m}$, thus it is well-posed. In contrast, we show that the initial value problem on higher codimension hypersurfaces is ill-posed, at least when specifying a finite number of derivatives of the data, due to the failure of uniqueness. This is in contrast to a uniqueness result which Courant and Hilbert deduce from Asgeirsson's mean value theorem, for which we give an independent derivation. The proofs use Fourier synthesis and the Holmgren-John uniqueness theorem.