Damped waves with singular damping on manifolds

We will discuss a new damped wave semigroup for damping exhibiting H\”{o}lder-type blowup near a hypersurface of a compact manifold. We will use this semigroup to prove a sharp energy decay result for singular damping on the torus, where the optimal rate of energy decay explicitly depends on the singularity of the damping. We also show that no finite time extinction could happen under this setting. This is a joint work with Perry Kleinhenz.