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Instability results for the damped wave equation in unbounded domains. (English) Zbl 1075.35018
Summary: We extend some previous results for the damped wave equation in bounded domains in \(\mathbb R^d\) to the unbounded case. In particular, we show that if the damping term is of the form \(\alpha a\) with bounded \(a\) taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive \(\alpha\).
In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrödinger operators to the unbounded case, which we believe to be of interest in their own right.

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B35 Stability in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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