Vol. 7, No. 1, 2014

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Sharp polynomial decay rates for the damped wave equation on the torus

Nalini Anantharaman and Matthieu Léautaud

Appendix: Stéphane Nonnenmacher

Vol. 7 (2014), No. 1, 159–214

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.

damped wave equation, polynomial decay, observability, Schrödinger group, torus, two-microlocal semiclassical measures, spectrum of the damped wave operator.
Mathematical Subject Classification 2010
Primary: 35A21, 35B35, 35L05, 35P20, 35S05
Secondary: 35B37, 93C20
Received: 11 October 2012
Revised: 21 May 2013
Accepted: 23 July 2013
Published: 7 May 2014
Nalini Anantharaman
Laboratoire de Mathématiques
Université Paris-Sud 11
Bâtiment 425
91405 Orsay Cedex
Matthieu Léautaud
Département de Mathématiques
Université Paris-Sud 11
Bâtiment 425
91405 Orsay Cedex
Université Paris Diderot–Paris 7
Bâtiment Sophie-Germain
75205 Paris Cedex 13
Stéphane Nonnenmacher
Institut de Physique Théorique
Unité de recherche associée au CNRS,CEA/Saclay
91191 Gif-sur-Yvette