On determinism and well-posedness in multiple time dimensions

@article{Craig2009OnDA,
  title={On determinism and well-posedness in multiple time dimensions},
  author={Walter Craig and Steven Weinstein},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2009},
  volume={465},
  pages={3023 - 3046}
}
  • W. Craig, Steven Weinstein
  • Published 1 December 2008
  • Mathematics
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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… 
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References

SHOWING 1-10 OF 16 REFERENCES
Contour integrals for the ultrahyperbolic wave equation
  • N. Woodhouse
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1992
The Penrose transform for the ultrahyperbolic wave equation is related to the problem of specifying data on an initial spacelike 2-plane. It is shown that a smooth solution can always be represented
Germ of a synthesis: space–time is spinorial, extra dimensions are time-like
  • G. Sparling
  • Physics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2007
A pressing issue for modern physics is the possibility of extra dimensions of space–time. Here, a novel approach to this question is put forward, with three facets: First, an integral transform is
LETTER TO THE EDITOR: On the dimensionality of spacetime
Some superstring theories have more than one effective low-energy limit corresponding to classical spacetimes with different dimensionalities. We argue that all but the (3 + 1)-dimensional one might
Survey of two-time physics
Two-time physics (2T) is a general reformulation of one-time physics (1T) that displays previously unnoticed hidden symmetries in 1T dynamical systems and establishes previously unknown duality-type
PARTIAL DIFFERENTIAL EQUATIONS
Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear
Duality and Strings, Space and Time
Duality symmetries in M--theory and string theory are reviewed, with particular emphasis on the way in which string winding modes and brane wrapping modes can lead to new spatial dimensions. Brane
Partial differential equations
Many physical problems involve quantities that depend on more than one variable. The temperature within a “large”1 solid body of conducting material varies with both time and location within the
Methods of Mathematical Physics, vol. II. Partial Differential Equations
Partial Differential Equations Fourth Edition
  • 1982
...
...