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In this paper, we propose a Cauchy type problem to the timelike Lorentzianeikonal equation on a globally hyperbolic space-time. For this equation, as thevalue of the solution on a Cauchy surface is known, we prove the existence ofviscosity solutions on the past set (future set) of the Cauchy surface.Furthermore, when the time orientation of viscosity solution is consistent, theuniqueness and stability of viscosity solutions are also obtained.

Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation