Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 1 (2014), 159-214.

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

Nalini Anantharaman and Matthieu Léautaud

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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.

Article information

Anal. PDE, Volume 7, Number 1 (2014), 159-214.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A21: Propagation of singularities 35B35: Stability 35L05: Wave equation 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 35S05: Pseudodifferential operators
Secondary: 35B37 93C20: Systems governed by partial differential equations

damped wave equation polynomial decay observability Schrödinger group torus two-microlocal semiclassical measures spectrum of the damped wave operator.


Anantharaman, Nalini; Léautaud, Matthieu. Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7 (2014), no. 1, 159--214. doi:10.2140/apde.2014.7.159.

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