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mourre’s commutators method

Journal of the Institute of Mathematics of Jussieu (1)

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LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE
Part of
Julien Royer
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 3 / June 2018

Published online by Cambridge University Press:

01 April 2016, pp. 509-540

Print publication:

June 2018
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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.