  We investigate the behavior of the solutions of a class of certain strictly
hyperbolic equations defined on $[0,T]\times \R^n$ 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 $x$ and with their $t$-derivative of order $\textnormal{O}(t^{-q}),$
where $q \in \big[1,\frac{3}{2}\big)$. For this purpose, an appropriate
generalized symbol class based on the metric is defined and the associated
Planck function is used to define an infinite order operator to perform
conjugation. We demonstrate that the solution not only experiences a loss of
regularity (usually observed for the case of coefficients bounded in $x$) but
also a decay in relation to the initial datum defined in a Sobolev space
tailored to the generalized symbol class. Further, we observe that a precise
behavior of the solution could be obtained by making an optimal choice of the
metric in relation to the coefficients of the given equation. We also derive
the cone conditions in the global setting.
