The influence of oscillations on energy estimates for damped wave models with time-dependent propagation speed and dissipation
| Autor: | Aslan, Halit Sevki, Reissig, Michael |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Modern Methods in Operator Theory and Harmonic Analysis, eds. A. Karapetyants, V. Kravchenko and E. Liflyand, (Springer Proceedings in Mathematics & Statistics, 2019) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/978-3-030-26748-3_17 |
| Popis: | 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. Comment: 37 pages, 2 figures |
| Databáze: | arXiv |
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