The Cauchy Problem for Symmetric Hyperbolic Systems with Nonlocal Potentials

TL;DR

This paper establishes well-posedness results for symmetric hyperbolic systems with nonlocal potentials on Lorentzian manifolds, including applications to Maxwell and Dirac equations, and discusses conditions for existence and non-existence of solutions.

Contribution

It provides new conditions under which the Cauchy problem for hyperbolic systems with nonlocal potentials is well-posed, including cases with retarded and short-range potentials.

Findings

Well-posedness established for retarded, bounded potentials.

Existence of solutions shown for short-range potentials under smallness conditions.

Counterexample demonstrates failure of solutions when potential bounds are too large.

Abstract

In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly bounded in time, we establish well-posedness of the Cauchy problem on a time strip, proving existence, uniqueness, and regularity of solutions. If the potential is not retarded but has only short time range, we show that strong solutions still exist, under the additional assumptions that the uniform bound in time is sufficiently small compared to the range in time and that its kernel decays sufficiently fast in time with respect to the zero-order terms of the system. Furthermore, we present a counterexample demonstrating that when the uniform bound is too large compared to the time range, solutions may fail to exist. As an application, we discuss Maxwell's equations in linear dispersive media on ultrastatic spacetimes, as well as the Dirac equation with nonlocal potential naturally arising in the theory of causal fermion systems. Our paper aims to represent the starting point for a rigorous study for the Cauchy problem for the semiclassical Einstein equations.