Resolvent estimates and local energy decay for hyperbolic equations

Abstract

We examine the cut-off resolvent R χ (λ) = χ (–Δ D – λ2)–1χ, where Δ D is the Laplacian with Dirichlet boundary condition and $\chi \in C_0^{\infty}(\mathbb{R}^n)$ equal to 1 in a neighborhood of the obstacle K. We show that if R χ (λ) has no poles for $\Im \lambda \geq -\delta,\: \delta > 0$ , then $\|R_{\chi}(\lambda)\|_{L^2 \to L^2} \leq C|\lambda|^{n-2},\: \lambda \in \mathbb{R},\: |\lambda| \geq C_0.$ This estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray. Keywords: Trapping obstacles, Resonances, Local energy decay, Cut-off resolvent