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
Schr\"odinger equation. We prove in an abstract setting that the observability
of the Schr\"odinger group implies that the semigroup associated to the damped
wave equation decays at rate $1/\sqrt{t}$ (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 $1/t$, as soon as the damping region does not satisfy GCC.
Conversely, for smooth damping coefficients $b$, we show that the semigroup
decays at rate $1/t^{1-\eps}$, for all $\eps >0$. 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), St\'{e}phane Nonnenmacher computes in an appendix
part of the spectrum of the associated damped wave operator, proving that the
semigroup cannot decay faster than $1/t^{2/3}$. In particular, our study shows
that the decay rate highly depends on the way $b$ vanishes.Comment: The paper, authored by N. Anantharaman and M. L\'eautaud, includes an
  appendix by S. Nonnenmache

Anantharaman, Nalini

Léautaud, Matthieu

Nonnenmacher, Stéphane

English

arXiv

The paper, authored by N. Anantharaman and M. Léautaud, includes an appendix by S.NonnenmacherWe 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 1$\sqrt{t}$ (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 1/$t$, 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 1/$t^{1−\epsilon}$ , for all $\epsilon$ > 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 1/$t^{2/3}$. In particular, our study emphasizes that the decay rate highly depends on the way $b$ vanishe

Decay rates for the damped wave equation on the torus

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