Asymptotics and Scattering for Critically Weakly Hyperbolic and Singular Systems

TL;DR

This paper analyzes a broad class of first-order hyperbolic systems with critical degeneracies and singularities at a specific time, providing detailed asymptotics, scattering solutions, and regularity loss quantifications, with applications to physics and cosmology.

Contribution

It extends existing weakly hyperbolic theory to include critically singular coefficients and offers precise asymptotic and scattering results for these systems.

Findings

Derived asymptotics of solutions near the singular time

Established scattering results with asymptotic data at the singularity

Quantified the regularity loss due to degeneracies

Abstract

We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain lower-order coefficients that blow up at a single time $t = 0$. In "critical" weakly hyperbolic settings, it is well-known that solutions lose a finite amount of regularity at the degenerate time $t = 0$. In this paper, we both improve upon the results in the weakly hyperbolic setting, and we extend this analysis to systems containing critically singular coefficients, which may also exhibit significantly modified asymptotics at $t = 0$. In particular, we give precise quantifications for (1) the asymptotics of solutions as $t$ approaches $0$; (2) the scattering problem of solving the system with asymptotic data at $t = 0$; and (3) the loss of regularity due to the degeneracies at $t = 0$. Finally, we discuss a variety of applications for these results, including to weakly hyperbolic and singular wave equations, equations of higher order, and equations arising from relativity and cosmology, e.g. at big bang singularities.