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

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}-\lambda ^2(t)\omega ^2(t)\varDelta u +\rho (t)\omega (t)u_t=0, &{} (t,x)\in [0,\infty )\times \mathbb {R}^n, \\ u(0,x)=u_0(x), \,\,\,\, u_t(0,x)=u_1(x), &{} x\in \mathbb {R}^n, \end{array}\right. } \end{aligned}$$

The coefficients $\lambda =\lambda (t)$ and $\rho =\rho (t)$ are shape functions and $\omega =\omega (t)$ is a bounded 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 Bui and Reissig (Fourier analysis, trends in mathematics. Birkhäuser, Basel, [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 $\omega =\omega (t)$ will influence energy estimates.