Second microlocalization and stabilization of damped wave equations on tori

We consider the question of stabilization for the damped wave equation on tori
$$(\partial_t^2 -\Delta )u +a(x) \partial _t u =0.$$
When the damping coefficient $a(x)$ is continuous the question is quite well understood and the geometric control condition is necessary and sufficient for uniform (hence exponential) decay to hold. When $a(x)$ is only $L^{\infty}$ there are still gaps in the understanding.
Using second microlocalization we completely solve the question for 
Damping coefficients of the form
$$a(x)=\sum_{i=1}^{J} a_j 1_{x\in R_j},$$
Where $R_j$ are cubes.
This is a joint work with P. Gérard

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