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

Abstract

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