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2014 Sharp polynomial decay rates for the damped wave equation on the torus
Nalini Anantharaman, Matthieu Léautaud
Anal. PDE 7(1): 159-214 (2014). DOI: 10.2140/apde.2014.7.159

Abstract

We address the decay rates of the energy for the damped wave equation when the damping coefficient b does not satisfy the geometric control condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger equation implies that the solutions of the damped wave equation decay at least like 1t (which is a stronger rate than the general logarithmic one predicted by the Lebeau theorem).

Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is 1t, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients b vanishing flatly enough, we show that the semigroup decays at least like 1t1ε, for all ε>0. The proof relies on a second microlocalization around trapped directions, and resolvent estimates.

In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), Stéphane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than 1t23. In particular, our study emphasizes that the decay rate highly depends on the way b vanishes.

Citation

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Nalini Anantharaman. Matthieu Léautaud. "Sharp polynomial decay rates for the damped wave equation on the torus." Anal. PDE 7 (1) 159 - 214, 2014. https://doi.org/10.2140/apde.2014.7.159

Information

Received: 11 October 2012; Revised: 21 May 2013; Accepted: 23 July 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1295.35075
MathSciNet: MR3219503
Digital Object Identifier: 10.2140/apde.2014.7.159

Subjects:
Primary: 35A21 , 35B35 , 35L05 , 35P20 , 35S05
Secondary: 35B37 , 93C20

Keywords: damped wave equation , observability , polynomial decay , Schrödinger group , spectrum of the damped wave operator. , Torus , two-microlocal semiclassical measures

Rights: Copyright © 2014 Mathematical Sciences Publishers

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