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We address the decay rates of the energy for the damped wave equation whenthe damping coefficient $b$ does not satisfy the Geometric Control Condition(GCC). First, we give a link with the controllability of the associatedSchr\"odinger equation. We prove in an abstract setting that the observabilityof the Schr\"odinger group implies that the semigroup associated to the dampedwave equation decays at rate $1/\sqrt{t}$ (which is a stronger rate than thegeneral logarithmic one predicted by the Lebeau Theorem).  Second, we focus on the 2-dimensional torus. We prove that the best decay onecan expect is $1/t$, as soon as the damping region does not satisfy GCC.Conversely, for smooth damping coefficients $b$, we show that the semigroupdecays at rate $1/t^{1-\eps}$, for all $\eps >0$. The proof relies on a secondmicrolocalization around trapped directions, and resolvent estimates.  In the case where the damping coefficient is a characteristic function of astrip (hence discontinuous), St\'{e}phane Nonnenmacher computes in an appendixpart of the spectrum of the associated damped wave operator, proving that thesemigroup cannot decay faster than $1/t^{2/3}$. In particular, our study showsthat the decay rate highly depends on the way $b$ vanishes.

Decay rates for the damped wave equation on the torus