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Asymptotic expansion of solutions to the wave equation with space-dependent damping

Motohiro Sobajima, Yuta Wakasugi

Abstract

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy method with suitable supersolutions of the corresponding parabolic problem.

Asymptotic expansion of solutions to the wave equation with space-dependent damping

Abstract

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy method with suitable supersolutions of the corresponding parabolic problem.
Paper Structure (18 sections, 25 theorems, 159 equations)

This paper contains 18 sections, 25 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Problem and backgrounds
  3. Main result
  4. A rough descripsion of strategy
  5. Construction of the paper
  6. Notations
  7. Preliminaries
  8. Weight functions
  9. Justification of the decomposition
  10. Well-posedness and regularity of solutions for the damped wave equation
  11. Regularity of solutions for the corresponding heat equation
  12. A decomposition lemma
  13. Derivation of the asymptotic expansion
  14. Energy estimates for the heat equation
  15. Energy estimates for the damped wave equation
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n$ be a nonnegative integer. Assume a for some $\alpha \in [0,1)$ and $a_0>0$. If $n+1 < \frac{N-\alpha}{2\alpha}$ and $\lambda \in [\frac{2\alpha}{2-\alpha}(n+1), \frac{N-\alpha}{2-\alpha})$, then there exist a positive integer $s = s(n)$ and a constant $m = m(n,\alpha,\lambda) >0$ such that t Then there exist profiles $\widetilde{V}_1 \ldots, \widetilde{V}_n\in C([0,\infty);L^2(\Omega))$ an

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1: SoWa17_AIMS, SoWa21_JMSJ
  • Remark 2.2
  • Definition 2.3: Kummer's confluent hypergeometric functions
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 41 more