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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.

##### MSC:
 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
##### Keywords:
indefinite damping; Schrödinger operators
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##### References:
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