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 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 $b$ vanishes.