Global Well-Posedness of a Class of Strictly Hyperbolic Cauchy Problems with Coefficients Non-Absolutely Continuous in Time

TL;DR

This paper establishes the global well-posedness, regularity, and decay properties of solutions to a class of strictly hyperbolic equations with time-dependent coefficients that are not absolutely continuous, using advanced phase space metrics.

Contribution

It introduces a generalized symbol class and an associated operator framework to analyze hyperbolic equations with non-absolutely continuous coefficients, providing new insights into solution behavior.

Findings

Solutions exhibit regularity loss and decay depending on initial data.

Optimal metric choice influences the precise solution behavior.

Cone conditions are derived for the global setting.

Abstract

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $[0,T]\times \R^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ and with their $t$-derivative of order $\textnormal{O}(t^{-q}),$ where $q \in \big[1,\frac{3}{2}\big)$. For this purpose, an appropriate generalized symbol class based on the metric is defined and the associated Planck function is used to define an infinite order operator to perform conjugation. We demonstrate that the solution not only experiences a loss of regularity (usually observed for the case of coefficients bounded in $x$) but also a decay in relation to the initial datum defined in a Sobolev space tailored to the generalized symbol class. Further, we observe that a precise behavior of the solution could be obtained by making an optimal choice of the metric in relation to the coefficients of the given equation. We also derive the cone conditions in the global setting.