Energy Estimates and Global Well-posedness for a Broad Class of Strictly Hyperbolic Cauchy Problems with Coefficients Singular in Time

TL;DR

This paper proves global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients that are singular in time and grow polynomially in space, using energy estimates and pseudodifferential conjugation techniques.

Contribution

It introduces a novel conjugation method with a loss operator to handle coefficients with singular behavior in time for hyperbolic equations.

Findings

Established energy estimates with variable loss depending on the conjugation order.

Proved global well-posedness under singular coefficient conditions.

Provided a counterexample and anisotropic cone conditions.

Abstract

The goal of this paper is to establish a global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients in $C^2((0,T];C^\infty(\mathbb{R}^n))$ growing polynomially in $x$ and singular in $t$. The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behavior is captured by the blow-up of the first and second $t$-derivatives of the coefficients which allows the coefficients to be either logarithmic-type or oscillatory-type near $t=0$. To arrive at an energy estimate, we perform a conjugation by a pseudodifferential operator of the form $e^{\nu(t)\Theta(x,D_x)},$ where $\Theta(x,D_x)$ explains the quantity of the loss by linking it to the metric on the phase space and the singular behavior while $\nu(t)$ gives a scale for the loss. We call the conjugating operator as {\itshape{loss operator}} and depending on its order we report that the solution experiences zero, arbitrarily small, finite or infinite loss in relation to the initial datum. We also provide a counterexample and derive the anisotropic cone conditions in our setting.