On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

TL;DR

This paper investigates how the spatial analyticity radius of solutions to semi-linear symmetric hyperbolic systems propagates over time, providing explicit lower bounds and extending results to Schrödinger equations with real-analytic potentials.

Contribution

It establishes precise lower bounds on the analyticity radius for solutions of hyperbolic systems and Schrödinger equations, advancing understanding of regularity propagation in these PDEs.

Findings

Analytic regularity propagates with explicit radius bounds over time.

Results apply to semi-linear symmetric hyperbolic systems and Schrödinger equations.

Provides lower bounds on the spatial analyticity radius as solutions evolve.

Abstract

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) \epsilon_0, the same happens for the solution u(t,.) for a certain radius \epsilon(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity \epsilon(t) as t grows. We also get similar results for the Schroedinger equation with a real-analytic electromagnetic potential.