In this talk, we will discuss the decay rates of damped waves when the Bardos-Lebeau-Rauch-Taylor Geometric Control Condition (GCC) fails, roughly speaking "there is a geodesic that does not enter the damped region".

Since decay rates are related to resolvent growth (Batty-Duyckaerts), we provide  and present a result of global lower resolvent bound, as well as applications of it in various settings.