We prove polynomial energy decay for polynomially controlled singular damping
    on the torus. This decay rate extends a result for bounded damping. We
    construct a semigroup that provides energy decay information in the singular
    case and use it to reduce the problem to resolvent estimates for the stationary
    damped wave equation. We then prove sufficiently good resolvent estimates using
    a version of the Morawetz multiplier method. We also establish exponential
    energy decay for such dampings on the circle, demonstrating that overdamping
    does not occur.



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