  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.
