Generalised energy conservation law for the wave equations with variable propagation speed

TL;DR

This paper studies the long-term energy behavior of solutions to wave equations with variable propagation speed, introducing a novel approach that combines higher derivative estimates with stabilization techniques.

Contribution

It presents a new method for analyzing the energy conservation law in wave equations with variable speed, integrating higher order derivative estimates and stabilization.

Findings

Established a generalized energy conservation law.

Demonstrated long-term energy stability under variable speed conditions.

Provided new estimates for higher order derivatives of coefficients.

Abstract

We investigate the long time behaviour of the $L^2$-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.