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

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 group implies that the semigroup associated to the damped wave equation decays at rate 1/ √ 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−ε , 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 1/t 2/3 . In particular, our study shows that the decay rate highly depends on the way b vanishes. KeywordsDamped wave equation, polynomial decay, observability, Schrödinger group, torus, two-microlocal semiclassical measures, spectrum of the damped wave operator.

Energy decay for a locally undamped wave equation

“…Conversely, if there is a geodesic that never meets supp(b), then uniform decay does not hold (see for instance [36]). In the case b ∈ C 0 (M ), the situation is simpler since uniform decay is equivalent to the fact that (1) Remark that the equality may fail, taking for instance b = 1 K where K is a compact Cantor set with positive measure satisfyingK = ∅, in which case supp(b) = K and ω b = ∅.…”

“…Decay rates for the damped wave equation on a flat metric with a lack of (GCC) have already been studied in [1,10,31,34]. In [1] it is proved that, on M = T n , decay at a rate t −1/2 always occurs if ω b = ∅.…”

“…In [1] it is proved that, on M = T n , decay at a rate t −1/2 always occurs if ω b = ∅. On the other hand, the decay cannot be better than t −1 as soon as (GCC) is strongly violated, i.e.…”

“…The energy of a solution is defined by (1) In the usual case where b is continuous, we have ω b = {b > 0} and ω b = supp b. As soon as ω b = ∅ one has E(u(t)) → 0 as t → +∞ (see for instance [28,29]).…”

“…Energy decay for a locally undamped wave equation ( * ) Matthieu Léautaud (1) and Nicolas Lerner (2) ABSTRACT. -We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated.…”