Energy estimates for wave equations with time dependent propagation speeds in the Gevrey class

The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but this is not true in general for time dependent propagation speeds. Indeed, it is considered in Hirosawa (2007) [3] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in Cm category. The main purpose of this paper is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates.