Sharp Decay Rate for the Damped Wave Equation With Convex-Shaped Damping

Abstract

Abstract We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. It was shown in [1] that, for sufficiently regular damping, the damped wave equation is stale at a rate sufficiently close to $t^{-1}$. We show that if the damping vanishes like a Hölder function $|x|^{\beta }$, and in addition, the boundary of the damped region is locally strictly convex with positive curvature, the wave is stable at rate $t^{-1+\frac {2}{2\beta +7}}$, which is better than the known optimal decay rate $t^{-1+\frac {1}{\beta +3}}$ for strip-shaped dampings of the same Hölder regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the sharp energy decay rate depends not only on the order of vanishing of the damping but also on the shape of the damped region. The main ingredient of the proof is the averaging method (normal form reduction) developed by Hitrik and Sjöstrand ([20], [35]).