  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.
