Stabilization rates for the damped wave equation with H\"older-regular damping

We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. We consider damping coefficients which are $0$ on a strip and vanish like polynomials, $x^{\beta}$. We prove that the semigroup cannot be stable at rate faster than $1/t^{(\beta+2)/(\beta+3)}$ by producing quasimodes of the associated stationary damped wave equation. We also prove that the semigroup is stable at rate at least as fast as $1/t^{(\beta+2)/(\beta+4)}$. These two results establish an explicit relation between the rate of vanishing of the damping and rate of decay of solutions. Our result partially generalizes a decay result of Nonnemacher in which the damping is an indicator function on a strip.