Second Microlocalization and Stabilization of Damped Wave Equations on Tori

Abstract

In this talk we present some recent results obtained in collaboration with P. Gérard (Stabilization of wave equations with rough dampings, 2016, in preparation) on the damped wave equation on two dimensional tori. With continuous dampings, the classical geometric control condition is necessary and sufficient for uniform stabilization. We give a very simple necessary and sufficient geometric condition on two dimensional tori for uniform stabilization in the special case where $$a(x) =\sum _{ j=1}^{N}a_{j}1_{x\in R_{j}}$$ (R j are rectangles, or more general polygons). The proof relies on second microlocalization.