Discrete time-dependent wave equations I. Semiclassical analysis

Abstract

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on the lattice $\hslash {\mathbb{Z}}^n$. 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 ${\ell }^2(\hslash {\mathbb{Z}}^n)$. However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter $\hslash \to 0$.