On determinism and well-posedness in multiple time dimensions

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.