For translation-invariant damping on the torus that vanishes like x to the beta power at the support boundary, the damped wave energy decays exactly like t to the minus (beta+2)/(beta+3).
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Sharp polynomial decay rates for the damped wave equation with H\"older-like damping
For translation-invariant damping on the torus that vanishes like x to the beta power at the support boundary, the damped wave energy decays exactly like t to the minus (beta+2)/(beta+3).