Global Well-Posedness of a Class of Hyperbolic Cauchy Problems with Coefficients Sublogarithmic in Time

TL;DR

This paper investigates the global well-posedness and regularity loss of solutions to a class of hyperbolic equations with coefficients that grow mildly in time, employing phase space metrics and pseudodifferential conjugation techniques.

Contribution

It introduces a novel approach using phase space metrics and a loss operator to analyze regularity loss in hyperbolic equations with sublogarithmic coefficient growth.

Findings

Solutions exhibit arbitrarily small regularity loss.

The method effectively characterizes the cone of dependence.

Energy estimates are obtained via pseudodifferential conjugation.

Abstract

The goal of this paper is to study global well-posedness, cone of dependence and loss of regularity of the solutions to a class of strictly hyperbolic equations with coefficients displaying "mild" blow-up of sublogarithmic order - $|\ln t|^\gamma,\gamma\in(0,1).$ The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The coefficients are polynomially bound in $x$ with their $x$-derivatives and $t$-derivative of order $O(t^{-\delta}),\delta \in [0,1),$ and $O(t^{-1}|\ln t|^{\gamma-1}), \gamma\in(0,1),$ respectively. We employ the Planck function associated with the metric to subdivide the extended phase space and define appropriate generalized parameter dependent symbol classes. To arrive at an energy estimate, we perform a conjugation by a pseudodifferential operator. This operator explains the loss of regularity by linking it to the metric on the phase space and the singular behavior. We call the conjugating operator as loss operator. We report that the solution experiences an arbitrarily small loss in relation to the initial datum defined in the Sobolev space tailored to the loss operator.