  The goal of this paper is to study global well-posedness, cone of dependence
and loss of regularity of the solutions to a class of strictly hyperbolic
equations with coefficients displaying "mild" blow-up of sublogarithmic order -
$|\ln t|^\gamma,\gamma\in(0,1).$ The problems we study are of strictly
hyperbolic type with respect to a generic weight and a metric on the phase
space. The coefficients are polynomially bound in $x$ with their
$x$-derivatives and $t$-derivative of order $O(t^{-\delta}),\delta \in [0,1),$
and $O(t^{-1}|\ln t|^{\gamma-1}), \gamma\in(0,1),$ respectively. We employ the
Planck function associated with the metric to subdivide the extended phase
space and define appropriate generalized parameter dependent symbol classes. To
arrive at an energy estimate, we perform a conjugation by a pseudodifferential
operator. This operator explains the loss of regularity by linking it to the
metric on the phase space and the singular behavior. We call the conjugating
operator as loss operator. We report that the solution experiences an
arbitrarily small loss in relation to the initial datum defined in the Sobolev
space tailored to the loss operator.
