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, $\Vert (G-\hat{I}»{)}^{-1}\Vert$, is approximately constant as $|\hat{I}»|\to +\infty$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, ${\overline{\mathbb{C}}}_{-}:=\{\hat{I}»\in \mathbb{C}:\mathrm{Re}\hat{I}»\le 0\}$. Our proof rests on a precise asymptotic analysis of the norm of the inverse of $T(\hat{I}»)$, the quadratic operator associated with $G$.

Corrected and simplified examples of level curves of the resolvent of the quadratic operator at the end of Sub-section 4.2