Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semigroup decay for the wave equation with unbounded damping

Antonio Arnal, Borbala Gerhat, Julien Royer, Petr Siegl

Abstract

We study the damped wave equation with a damping coefficient which is possibly singular and unbounded at infinity. In general, zero belongs to the spectrum of the corresponding generator, which prevents a uniform (exponential) decay for the energy. However, for initial conditions in a suitable subspace, a detailed analysis of the resolvent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy.

Semigroup decay for the wave equation with unbounded damping

Abstract

We study the damped wave equation with a damping coefficient which is possibly singular and unbounded at infinity. In general, zero belongs to the spectrum of the corresponding generator, which prevents a uniform (exponential) decay for the energy. However, for initial conditions in a suitable subspace, a detailed analysis of the resolvent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy.
Paper Structure (22 sections, 34 theorems, 296 equations, 2 figures)

This paper contains 22 sections, 34 theorems, 296 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Generator of the damped wave equation for unbounded damping
  3. Definition of the generator and solution of the damped wave equation
  4. The Schur complement
  5. The resolvent of A
  6. Range of A and the space K
  7. Restriction of the semigroup to K
  8. Resolvent estimates for low frequencies
  9. Statements for the low frequency resolvent estimates
  10. The resolvent norm of A near zero
  11. Low frequency estimate for the associated quadratic form
  12. Resolvent estimates for high frequencies
  13. Time decay
  14. Proof of the main results
  15. From resolvent estimates to time decay
  16. ...and 7 more sections

Key Result

Proposition 1.1

Let $\Omega \subset \mathbb{R}^d$ be non-empty and open, and let $0\leq a,q \in {L^1_{\mathrm{loc}}(\Omega)}$. If there exist $\nu > 0$ and $C > 0$ such that for all $F=(f,g) \in {\mathcal{H}}$ and $t \geq 0$ the solution $u(t)$ of dwe.2ndorder satisfies then there exists $M > 0$ such that Moreover, if $a(x) \geq a_0 > 0$ a.e. in $\Omega$, then the reverse implication holds, so that eq:exp-decay

Figures (2)

  • Figure 2.1: Figure reproduced from Arifoski-2020-52. Numerical computation of spectrum (in black, see \ref{['sp.x^2']}) and pseudospectra ($\log_{10}$ scale, approximation by $800\times800$ matrix) of ${\mathcal{A}}$ (see Section \ref{['sec:dwe']}) with $a(x)=2x^2$ and $q(x)\equiv0$, $x\in {\mathbb{R}}$.
  • Figure 5.1: The contours $\Gamma_R$, $\Gamma_{R,\pm}$, $\Gamma_{R,\varepsilon}$, $\Gamma_{R,{\mathsf{high}}}$ and $\Gamma_{\varepsilon,{\mathsf{low}}}$(in grey a region which contains the spectrum of $A$).

Theorems & Definitions (67)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1: gerhat2024schur
  • Remark 2.2
  • Theorem 2.3: gerhat2024schur, Freitas-2018-264
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 57 more