Discrete time-dependent wave equations I. Semiclassical analysis

TL;DR

This paper studies semiclassical wave equations with time-dependent speeds on a lattice, showing well-posedness despite singularities, and explores limits in Gevrey and Sobolev spaces as the semiclassical parameter approaches zero.

Contribution

It introduces a novel analysis of hyperbolic equations with vanishing speeds on lattices, demonstrating well-posedness in discrete and certain functional spaces.

Findings

Well-posedness in ll^2(\u03b7 ext{Z}^n) despite singular speeds

Recovery of well-posedness in Gevrey and Sobolev spaces as ar

Contrasts with Euclidean case where problems are ill-posed

Abstract

In this paper we consider a semiclassical version of the wave equations with singular H\"{o}lder time-dependent propagation speeds on the lattice $\hbar\mathbb{Z}^{n}$. We allow the propagation speed to vanish leading to the weakly hyperbolic nature of the equations. Curiously, very much contrary to the Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] and by other authors, the Cauchy problem, in this case, is well-posed in $\ell^2(\hbar\mathbb{Z}^{n})$. However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter $\hbar\to 0$.