Discrete time-dependent wave equations I. Semiclassical analysis

In this paper we consider a semiclassical version of the wave equations with singular Holder time dependent propagation speeds on the lattice hZ(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 l(2)(hZ(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 h -> 0. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we study a semiclassical version of wave equations with singular Holder time dependent propagation speeds on a lattice. Contrary to previous studies, we find that the Cauchy problem is well-posed in certain spaces and recover the well-posedness results as the semiclassical parameter tends to zero.