Some local questions for hyperbolic systems with non-regular time dependent coefficients

TL;DR

This paper establishes finite propagation speed and local well-posedness for microlocally symmetrizable hyperbolic systems with time-dependent coefficients using Fourier analysis, under mild regularity assumptions.

Contribution

It provides new results on finite propagation speed and local existence for hyperbolic systems with purely time-dependent coefficients, using Fourier transform methods.

Findings

Finite propagation speed established with precise estimates.

Local existence and uniqueness proven under mild regularity.

Method relies on Fourier transform, limited to time-dependent coefficients.

Abstract

In this note we investigate local properties for microlocally symmetrizable hyperbolic systems with just time dependent coefficients. Thanks to Paley-Wiener theorem, we establish finite propagation speed by showing precise estimates on the evolution of the support of the solution in terms of suitable norms of the coefficients of the operator and of the symmetrizer. From this result, local existence and uniqueness follow by quite standard methods. Our argument relies on the use of Fourier transform, and it cannot be extended to operators whose coefficients depend also on the space variables. On the other hand, it works under very mild regularity assumptions on the coefficients of the operator and of the symmetrizer.