On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

TL;DR

This paper studies the well-posedness of the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundaries, establishing existence, uniqueness, and conditions for hyperbolic systems.

Contribution

It introduces admissible boundary conditions for Friedrichs systems on such manifolds and proves well-posedness and uniqueness results, extending classical theory to manifolds with boundary.

Findings

Existence and uniqueness of strong solutions under admissible boundary conditions

Well-posedness of the Cauchy problem for hyperbolic Friedrichs systems

Examples of Friedrichs systems with admissible boundary conditions

Abstract

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.