On determinism and wellposedness in multiple time dimensions
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 codimensionone hypersurfaces has global unique solutions in the Sobolev spaces $H^{m}$, thus it is wellposed. In contrast, we show that the initial value problem on higher codimension hypersurfaces is illposed, 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 HolmgrenJohn uniqueness theorem.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 July 2009
 DOI:
 10.1098/rspa.2009.0097
 arXiv:
 arXiv:0812.0210
 Bibcode:
 2009RSPSA.465.3023C
 Keywords:

 Mathematical Physics;
 35Q99
 EPrint:
 doi:10.1098/rspa.2009.0097