The damped wave equation with unbounded damping

Pedro Freitas*, Petr Siegl, Christiane Tretter

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)
184 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)7023-7054
JournalJournal of Differential Equations
Volume264
Issue number12
Early online date21 Feb 2018
DOIs
Publication statusPublished - 15 Jun 2018

Keywords

  • Damped wave equation
  • Essential spectrum
  • Quadratic operator function with unbounded coefficients
  • Schrödinger operators with complex potentials
  • Unbounded damping

ASJC Scopus subject areas

  • Analysis

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