The damped wave equation with unbounded damping

TL;DR

This paper studies the spectral and semigroup properties of the damped wave equation on unbounded domains with unbounded damping, revealing new phenomena like essential spectrum and eigenvalue convergence.

Contribution

It provides a comprehensive analysis of spectral behavior and semigroup generation for damped wave equations with unbounded damping, including effects on exponential stability.

Findings

Essential spectrum appears on the negative real axis.

Non-real eigenvalues converge in the diverging damping regime.

Presence of essential spectrum prevents exponential decay estimates.

Abstract

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.