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

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}-λ^2(t)ω^2(t)Δu +ρ(t)ω(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 $\omega (t)\equiv 1$ 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.

37 pages, 2 figures