Under smallness and decay conditions on nonlocal potentials, symmetric hyperbolic systems on curved spacetimes admit strong solutions to the Cauchy problem, with a sharp threshold beyond which solutions fail.
On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
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In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.
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The Cauchy Problem for Symmetric Hyperbolic Systems with Nonlocal Potentials
Under smallness and decay conditions on nonlocal potentials, symmetric hyperbolic systems on curved spacetimes admit strong solutions to the Cauchy problem, with a sharp threshold beyond which solutions fail.