  We investigate the behavior of the solutions of a class of certain strictly
hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ 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 $x$ with their $x$-derivatives and $t$-derivative of
order ${O}(t^{-\delta}),\delta \in [0,1),$ and ${O}(t^{-1})$ respectively. This
type of singular behavior allows coefficients to be either oscillatory or
logarithmically bounded at $t=0$. We use the Planck function associated with
the metric to subdivide the extended phase space and define an appropriate
generalized parameter dependent symbol class. We report that the solution not
only experiences a finite loss of derivatives but also a decay in relation to
the initial datum defined in a Sobolev space tailored to the metric. Our
analysis suggests that an infinite loss is quite expected when the order of
singularity of the first time derivative of the leading coefficients exceeds
$O(t^{-1})$. We confirm this by providing counterexamples. Further, using the
$L^1$ integrability of the logarithmic singularity in $t$ and the global
properties of the operator with respect to $x$, we derive the anisotropic cone
conditions in our setting.
