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

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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 equation* utt-λ²(t)ω²(t)Δu +ρ(t)ω(t)uₜ=0, u(0,x)=u₀(x), uₜ(0,x)=u₁(x). equation* The coefficients λ=λ(t) and ρ=ρ(t) are shape functions and ω=ω(t) is an oscillating function. If ω(t)≡1 and ρ(t)uₜ is an "effective" dissipation term, then L²-L² 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.