Discrete time-dependent wave equations I. Semiclassical analysis

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on lattice ħZn. We allow speed to vanish leading weakly hyperbolic nature equations. Curiously, very much contrary Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] other authors, Cauchy problem in is well-posed ℓ2(ħZn). However, also recover well-posedness results intersection certain Gevrey Sobolev spaces limit parameter ħ→0.