I will present new results in the study of damped-wave equations for systems close to being completely integrable. Precisely, we will consider sub-principal perturbations of semiclassical harmonic oscillators and damped Baouendi-Grushin operators. For these systems, we obtain microlocal resolvent estimates near the real axis when weak-geometric control conditions are assumed. We test some of these estimates via the construction of quasimodes within the damping region.

This talk is based on joint works with Gabriel Rivière, Fabricio Macià and Chenmin Sun.