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Optimal energy decay rates for Klein-Gordon equations with Kelvin-Voigt damping

Filippo Dell'Oro, Lassi Paunonen, David Seifert

Abstract

We study the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. One of the interesting features of the equation is that the generator of the associated $C_0$-semigroup has multiple spectral points on the imaginary axis. As our main result, we show that the energy of every possible solution converges to zero as time goes to infinity and, moreover, we provide an optimal polynomial energy decay rate for a certain class of solutions.

Optimal energy decay rates for Klein-Gordon equations with Kelvin-Voigt damping

Abstract

We study the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. One of the interesting features of the equation is that the generator of the associated -semigroup has multiple spectral points on the imaginary axis. As our main result, we show that the energy of every possible solution converges to zero as time goes to infinity and, moreover, we provide an optimal polynomial energy decay rate for a certain class of solutions.

Paper Structure

This paper contains 5 sections, 6 theorems, 50 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Semigroup Generator and Its Boundary Spectrum
  3. Resolvent Estimates
  4. Asymptotic Stability and Quantified Decay Rates
  5. The Half-Line Case

Key Result

Theorem 2.1

The following hold:

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['res-est']}
  • Theorem 4.1
  • proof
  • Proposition 4.2