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Stabilization of linear waves with inhomogeneous Neumann boundary conditions

Türker Özsarı, İdem Susuzlu

Abstract

We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilization of solutions with decay rates characterized by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Stabilization of linear waves with inhomogeneous Neumann boundary conditions

Abstract

We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilization of solutions with decay rates characterized by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Paper Structure

This paper contains 20 sections, 13 theorems, 170 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Mathematical models and notation
  3. Motivation and orientation
  4. Previous studies
  5. Dynamic extension for the wave equation
  6. Damped wave equation
  7. Homogenization
  8. Existence and uniqueness
  9. Uniform Stabilization
  10. Viscoelastic wave equation
  11. Homogenization
  12. Existence and uniqueness
  13. Uniform stabilization
  14. Numerical experiments for 1D wave equation with a distributed damping
  15. Numerical Solution
  16. ...and 5 more sections

Key Result

Lemma 2.1

$E:(h,u^0,u^1)\rightarrow (v,v_t)$ is continuous from into

Figures (8)

  • Figure 1: Linear waves subject to external Neumann manipulation and spatially decaying damping.
  • Figure 2: Linear waves subject to external Neumann manipulation and periodic damping.
  • Figure 3: Linear waves subject to external Neumann manipulation and quadratic damping.
  • Figure 4: Classical wave solution, $g(t)=h(t)=0$.
  • Figure 5: Viscoelastic waves subject to Neumann manipulation and exponentially decaying relaxation.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Remark 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 18 more