In this talk, I will present new results concerning the study of the resolvent of the damped-wave operator associated with the sub-elliptic Laplacian known as Baouendi-Grushin operator on the two-dimensional flat torus. From different hypothesis on the geometry of the damping region and the Hölder regularity of the damping term, I will show sharp resolvent estimates of the associated non-selfadjoint operator on the real axis. As an application, sharp energy-decay-rates of the damped-wave equation are obtained. The proofs are based on the study of two-microlocal semiclassical measures, normal form reductions and constructions of quasimodes in different parts of the phase-space.
This work has been done in collaboration with Chenmin Sun. Reference: arXiv:2201.08189.