# Équations des ondes avec des perturbations dépendantes du temps

Abstract : We study the wave equation $\partial_t^2 u-\Div_x(a(t,x)\nabla_xu)=0$ with time-periodic and scalar metric $a(t,x)$ equal to $1$ outside a compact set with respect to $x$. Our goal is to estimate the solutions of this equation with initial data lying in the energy space $\dot{H}^1({\R}^n)\times L^2({\R}^n)$. More precisely, we will establish global Strichartz estimates as well as local energy decay under some assumptions. We will distinguish the case of odd dimensions from the case of even dimensions. Set $n$ the space dimension. In the first part of our work, we deal with odd dimensions $n\geq3$. We assume that $a(t,x)$ is non-trapping and that there is no resonance $z\in\mathbb{C}$ with modulus greater than $1$. We show that this assumptions imply local energy decay and $L^2$ integrability with respect to time of the local energy. Then, we establish local Strichartz estimates for the solutions of $\partial_t^2 u-\Div_x(a(t,x)\nabla_xu)=0$. Combining these arguments, we obtain global Strichartz estimates. In the second part of our work, we deal with even dimensions $n\geq4$. Consider the cut-off resolvent $R_\chi(\theta)=\chi(\mathcal U(T)-e^{-i\theta})^{-1}\chi$ , where $\chi\in{\CI}$, $T$ is the period of $a(t,x)$ and $\mathcal U(T)$ is the propagator associate to the equation at time $T$. We assume that $a(t,x)$ is non-trapping and the cut-off resolvent $R_\chi(\theta)$ admit an analytic continuation to $\{\theta\in\mathbb{C}\ :\ \textrm{Im}(\theta) \geq 0\}$, for $n \geq 3$, odd, and to \\ $\{ \theta\in\mathbb C\ :\ \textrm{Im}(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,\ k\in\mathbb{Z},\ \mu\geq0\}$ for $n \geq4$, even. Moreover, for $n \geq4$ even, we assume that $R_\chi(\theta)$ is bounded in some neighborhood of $\theta=0$. We prove that these assumptions imply local energy decay. Combining this argument with the results of the first part, we obtain global Strichartz estimates for any dimensions $n\geq3$.
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https://tel.archives-ouvertes.fr/tel-00576179
Contributor : Yavar Kian <>
Submitted on : Sunday, March 13, 2011 - 10:17:05 AM
Last modification on : Thursday, January 11, 2018 - 6:21:22 AM
Long-term archiving on: : Tuesday, June 14, 2011 - 2:25:44 AM

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• HAL Id : tel-00576179, version 1

### Citation

Yavar Kian. Équations des ondes avec des perturbations dépendantes du temps. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2010. Français. ⟨tel-00576179⟩

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