Global Well-posedness of a Class of Singular Hyperbolic Cauchy Problems

TL;DR

This paper proves the global well-posedness and analyzes the regularity loss for a class of singular hyperbolic equations with unbounded coefficients near a hyperplane, using phase space metrics and pseudodifferential operators.

Contribution

It introduces a novel framework involving phase space metrics and infinite order pseudodifferential operators to handle singular hyperbolic equations with unbounded coefficients.

Findings

Established global well-posedness for the class of equations.

Demonstrated cone condition and regularity loss.

Provided counterexamples illustrating limitations.

Abstract

The goal of this paper is to establish a global well-posedness, cone condition and loss of regularity for singular hyperbolic equations with coefficients in { $L^1((0,T];C^\infty(\mathbb{R}^n)) \cap C^1((0,T];C^\infty(\mathbb{R}^n))$} and Cauchy data in an appropriate Sobolev space tailored to a metric on the phase space. The coefficients are unbounded near the singular hyperplane $t=0$ and polynomially growing as $|x| \to \infty.$ The singular behavior is characterized by the blow-up rate of the coefficients and their first $t$-derivatives near $t=0.$ In order to study the interplay of the singularity in $t$ and unboundedness in $x$, we consider a class of metrics on the phase space. Our methodology relies on the use of the Planck function associated to the metric to subdivide the extended phase and to define an infinite order pseudodifferential operator for the conjugation. We also give some counterexamples.