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We study the generator $G$ of the one-dimensional damped wave equation withunbounded damping. We show that the norm of the corresponding resolventoperator, $\| (G - \lambda)^{-1} \|$, is approximately constant as $|\lambda|\to +\infty$ on vertical strips of bounded width contained in the closure ofthe left-hand side complex semi-plane, $\overline{\mathbb{C}}_{-} := \{\lambda\in \mathbb{C}: \operatorname{Re} \lambda \le 0\}$. Our proof rests on aprecise asymptotic analysis of the norm of the inverse of $T(\lambda)$, thequadratic operator associated with $G$.

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping