The damped wave equation with singular damping

Petr Siegl; Pedro Freitas; Nicolas Hefti

The damped wave equation with singular damping

Petr Siegl, Pedro Freitas, Nicolas Hefti

School of Mathematics and Physics

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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 language	English
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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The Damped Wave Equation with Singular Damping
© 2020 American Mathematical Society. This work is made available online in accordance with the publisher’s policies. Please refer to any applicable terms of use of the publisher.
Accepted author manuscript, 2.49 MBLicence: Unspecified

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Siegl, P., Freitas, P., & Hefti, N. (2020). The damped wave equation with singular damping. Proceedings of the American Mathematical Society.

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N2 - 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.

AB - 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.

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

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