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LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE

Part of: Hyperbolic equations and systems General theory of linear operators Partial differential equations Special classes of linear operators

Published online by Cambridge University Press:  01 April 2016

Julien Royer
Julien Royer*
Affiliation:
Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cédex 9, France (julien.royer@math.univ-toulouse.fr)
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Abstract

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$. The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

Keywords

local energy decaydamped wave equationMourre’s commutators methodnon-self-adjoint operatorsresolvent estimates

MSC classification

Primary: 35B40: Asymptotic behavior of solutions 35L05: Wave equation 47A40: Scattering theory 47A55: Perturbation theory 47B44: Accretive operators, dissipative operators, etc.
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 3 , June 2018 , pp. 509 - 540
Copyright
© Cambridge University Press 2016 

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