Sharp polynomial decay for polynomially singular damping on the torus

We study energy decay rates for the damped wave equation with unbounded damping, without the geometric control condition. Our main decay result is sharp polynomial energy decay for polynomially controlled singular damping on the torus. We also prove that for normally $L^p$-damping on compact manifolds, the Schr\"odinger observability gives $p$-dependent polynomial decay, and finite time extinction cannot occur. We show that polynomially controlled singular damping on the circle gives exponential decay.