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Optimal stabilization rate for the wave equation with hyperbolic boundary condition

Hugo Parada, Nicolas Vanspranghe

TL;DR

The paper analyzes energy decay for the wave equation on a bounded domain with a dynamic (hyperbolic) boundary condition and boundary damping. By establishing a robust operator-theoretic framework and performing detailed microlocal analysis of high-frequency quasimodes, it proves that the energy decays at a polynomial rate of $o(t^{-1})$ under geometric control on the dynamic boundary portion and a finite-contact condition. The authors further show sharpness of this rate via a disk example, using a spectral construction inspired by Bessel functions to demonstrate obstructions to faster decay. A decoupling argument and a boundary observability result underpin the resolvent estimates that drive the decay result, providing a comprehensive picture of stabilization limits for dynamic boundary systems. These results illuminate the influence of boundary dynamics and GCC on stabilization performance and guide design considerations for boundary damping in multi-dimensional domains.

Abstract

We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a possibly empty, disjoint part of the boundary may be kept at rest provided that the dynamic part satisfies the geometric control condition. We also prove that this decay rate is sharp. Our results follow from resolvent estimates, which we establish by studying high-frequency quasimodes.

Optimal stabilization rate for the wave equation with hyperbolic boundary condition

TL;DR

The paper analyzes energy decay for the wave equation on a bounded domain with a dynamic (hyperbolic) boundary condition and boundary damping. By establishing a robust operator-theoretic framework and performing detailed microlocal analysis of high-frequency quasimodes, it proves that the energy decays at a polynomial rate of under geometric control on the dynamic boundary portion and a finite-contact condition. The authors further show sharpness of this rate via a disk example, using a spectral construction inspired by Bessel functions to demonstrate obstructions to faster decay. A decoupling argument and a boundary observability result underpin the resolvent estimates that drive the decay result, providing a comprehensive picture of stabilization limits for dynamic boundary systems. These results illuminate the influence of boundary dynamics and GCC on stabilization performance and guide design considerations for boundary damping in multi-dimensional domains.

Abstract

We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a possibly empty, disjoint part of the boundary may be kept at rest provided that the dynamic part satisfies the geometric control condition. We also prove that this decay rate is sharp. Our results follow from resolvent estimates, which we establish by studying high-frequency quasimodes.
Paper Structure (20 sections, 28 theorems, 215 equations, 1 figure)

This paper contains 20 sections, 28 theorems, 215 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Background and literature review
  4. Preliminaries and operator model
  5. The Laplace--Beltrami operator on the boundary
  6. Elements of differential geometry
  7. Local setting near the boundary
  8. Well-posedness and asymptotic stability
  9. Functional setting and semigroup generation
  10. Energy and stability
  11. Energy decay rate and analysis of quasimodes
  12. Statement of the main result
  13. Analysis of quasimodes
  14. Preliminary estimates
  15. Microlocalized estimates near the boundary
  16. ...and 5 more sections

Key Result

Theorem 1.2

Suppose that as:dyn holds. Let $(u_0, u_1)\in H^{2}(\Omega)\times H^{1}(\Omega)$ be such that $(u_{0},u_{1})|_{\Gamma_0} \in H^{2}(\Gamma_0)\times H^{1}(\Gamma_0)$ and $(u_0, u_1)|_{\Gamma_1} = 0$, and let $u$ the unique solution of eq:IBVP with initial data $(u_0, u_1)$. Then,

Figures (1)

  • Figure 1: Two generalized bicharacteristics (projected onto the $x$-space), with different type of points of contact: nondiffractive ($N$), diffractive ($D$), gliding ($G$). Adapted from miller2002escape.

Theorems & Definitions (67)

  • Theorem 1.2: Energy decay rate
  • Theorem 1.3: Lower bound on decay, disk case
  • Proposition 2.1: Characterization of the domain
  • proof
  • Proposition 2.2: Semigroup generation
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 57 more