The damped wave equation with singular damping

Pedro Freitas; Nicolas Hefti; Petr Siegl

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arXiv · 2002.03440

The damped wave equation with singular damping

Pedro Freitas·Nicolas Hefti·Petr Siegl

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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 $\alpha/x$, $\alpha>0$. We establish the exponential stability of the semigroup for all positive $\alpha$, 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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Pedro Freitas, Nicolas Hefti, Petr Siegl. 2020-02-09. The damped wave equation with singular damping. https://arxiv.org/abs/2002.03440

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