Global Well-Posedness of a Class of Hyperbolic Cauchy Problems with Coefficients Sublogarithmic in Time

Abstract

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.