The influence of oscillations on energy estimates for damped wave models with time-dependent propagation speed and dissipation

TL;DR

This paper derives higher order energy estimates for damped wave equations with time-varying and oscillating coefficients, revealing how oscillations influence energy behavior in such models.

Contribution

It generalizes existing energy estimates to include oscillating coefficients, highlighting the impact of oscillations on energy estimates for damped wave models.

Findings

Oscillations significantly affect energy estimates.

Higher order energy estimates are established for oscillating coefficients.

The interplay between shape functions and oscillations influences energy behavior.

Abstract

The aim of this paper is to derive higher order energy estimates for solutions to the Cauchy problem for damped wave models with time-dependent propagation speed and dissipation. The model of interest is \begin{equation*} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u +\rho(t)\omega(t)u_t=0, \quad u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x). \end{equation*} The coefficients $\lambda=\lambda(t)$ and $\rho=\rho(t)$ are shape functions and $\omega=\omega(t)$ is an oscillating function. If $\omega(t)\equiv1$ and $\rho(t)u_t$ is an "effective" dissipation term, then $L^2-L^2$ energy estimates are proved in [2]. In contrast, the main goal of the present paper is to generalize the previous results to coefficients including an oscillating function in the time-dependent coefficients. We will explain how the interplay between the shape functions and oscillating behavior of the coefficient will influence energy estimates.