The damped wave equation with singular damping

Petr Siegl, Pedro Freitas, Nicolas Hefti

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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 languageEnglish
JournalProceedings of the American Mathematical Society
Early online date22 May 2020
Publication statusEarly online date - 22 May 2020

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