### Abstract

We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping
of the form α/x, α > 0. We establish the exponential stability of the semigroup for all positive α, and determine conditions for the spectrum to consist
of a finite number of eigenvalues. As a consequence, we fully characterize
the set of initial conditions for which there is extinction of solutions in finite
time. Finally, we propose two open problems related to extremal decay rates
of solutions.

Original language | English |
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Journal | Proceedings of the American Mathematical Society |

Early online date | 22 May 2020 |

Publication status | Early online date - 22 May 2020 |

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## Cite this

Siegl, P., Freitas, P., & Hefti, N. (2020). The damped wave equation with singular damping.

*Proceedings of the American Mathematical Society*.