Strictly Hyperbolic Cauchy Problems on $\mathbb{R}^n$ with Unbounded and Singular Coefficients

Abstract

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.