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

TL;DR

This paper analyzes the resolvent estimates for the one-dimensional damped wave equation with unbounded damping, showing that the resolvent norm remains approximately constant at high frequencies within certain complex plane regions.

Contribution

It provides a detailed asymptotic analysis of the resolvent operator for the damped wave generator with unbounded damping, revealing its behavior at high frequencies.

Findings

Resolvent norm remains approximately constant as frequency increases.

Asymptotic behavior of the inverse quadratic operator is characterized.

Results apply to complex semi-plane regions with bounded width.

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 $G$.