The damped wave equation with singular damping

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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