Generalized energy conservation for linear wave equations with time-dependent propagation speed

TL;DR

This paper studies wave equations with time-varying speeds, establishing conditions for energy conservation, deriving bounds for energy growth when conservation fails, and providing counterexamples to show the bounds' optimality.

Contribution

It introduces generalized energy conservation criteria for wave equations with oscillating propagation speeds and develops sharp bounds on energy growth, supported by counterexamples.

Findings

Conditions for generalized energy conservation established.

Derived upper bounds for energy growth when conservation fails.

Counterexamples demonstrate the optimality of bounds.

Abstract

We consider a wave equation with a time-dependent propagation speed, whose potential oscillations are controlled through bounds on its first and second derivatives and by limiting the integral of the difference with a fixed constant. We investigate when the wave equation exhibits generalized energy conservation (GEC), meaning that the energy of all solutions remains bounded for all times by a multiple of the initial energy. When GEC is not satisfied, we provide upper bounds for the growth of the energy. These upper bounds are derived by analyzing the growth of the Fourier components of the solution. Depending on the frequency and the time interval, different energy inequalities are employed to fully exploit our assumptions on the propagation speed. Finally, we present counterexamples that demonstrate the optimality of our upper bound estimates.