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<p>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 <i>L</i><sup><i>p</i></sup>-damping on compact manifolds, the Schrödinger observability gives <i>p</i>-dependent polynomial decay, and finite time extinction cannot occur. We show that polynomially controlled singular damping on the circle gives exponential decay.</p>

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Sharp Polynomial Decay for Polynomially Singular Damping on the Torus

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月24日

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  • Perry Kleinhenz,
  • Ruoyu P. T. Wang

摘要

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 Lp-damping on compact manifolds, the Schrödinger 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.