On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions (Q493792)

A Cauchy problem for the wave equation with time dependent propagation speed is investigated. Denote \(a(t)\) the propagation speed and \(E(t)\) the total energy defined as the sum of the elastic and kinetic energy. Under some special assumptions on the propagation speed \(a(t)\) and initial data, one proves lower and upper bounds for the energy \(E(t)\), in terms of \(E(0)\). There is a large discussion around the validity of different generalized energy conservation type inequalities, the GECs, as well as the energy estimates.