  We analyze new phenomena arising in linear damped wave equations on unbounded
domains when the damping is allowed to become unbounded at infinity. We prove
the generation of a contraction semigroup, study the relation between the
spectra of the semigroup generator and the associated quadratic operator
function, the convergence of non-real eigenvalues in the asymptotic regime of
diverging damping on a subdomain, and we investigate the appearance of
essential spectrum on the negative real axis. We further show that the presence
of the latter prevents exponential estimates for the semigroup and turns out to
be a robust effect that cannot be easily canceled by adding a positive
potential. These analytic results are illustrated by examples.
