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Increasing stability for inverse acoustic source problems in the time domain

Chun Liu, Suliang Si, Guanghui Hu, Bo Zhang

Abstract

This paper is concerned with inverse source problems for the acoustic wave equation in the full space R^3, where the source term is compactly supported in both time and spatial variables. The main goal is to investigate increasing stability for the wave equation in terms of the interval length of given parameters (e.g., bandwith of the temporal component of the source function). We establish increasing stability estimates of the L^2 -norm of the source function by using only the Dirichlet boundary data. Our method relies on the Huygens principle, the Fourier transform and explicit bounds for the continuation of analytic functions.

Increasing stability for inverse acoustic source problems in the time domain

Abstract

This paper is concerned with inverse source problems for the acoustic wave equation in the full space R^3, where the source term is compactly supported in both time and spatial variables. The main goal is to investigate increasing stability for the wave equation in terms of the interval length of given parameters (e.g., bandwith of the temporal component of the source function). We establish increasing stability estimates of the L^2 -norm of the source function by using only the Dirichlet boundary data. Our method relies on the Huygens principle, the Fourier transform and explicit bounds for the continuation of analytic functions.
Paper Structure (8 sections, 8 theorems, 143 equations, 7 figures)

This paper contains 8 sections, 8 theorems, 143 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Proofs of Theorem \ref{['TH1']}
  5. Proof of Theorem \ref{['TH3']}
  6. Proof of Theorem \ref{['TH4']}
  7. Conclusion
  8. Acknowledgment

Key Result

Theorem 2.1

Let the condition af hold and let $T>2R+T_0$. Assume that $g(t)$ is given and $\|f\|_{H^1({\mathbb R}^3)}\leq M$ where $M>1$ is a constant. Then where $\epsilon=\|u\|_{H^2([0,T];H^\frac{3}{2}(\partial B_R))}$.

Figures (7)

  • Figure 1: $E(s)$ is the shaded area, $|\xi|^2=\zeta^2$.
  • Figure 2: $E_1(s)$ is the shaded area, $|\xi|^2=\zeta^2$.
  • Figure 3: $E_2(s)$ is the shaded area, $|\xi|^2=\zeta^2$.
  • Figure 4: $E(s)$ is the shaded area.
  • Figure 5: $E_1(s)$ is the shaded area.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Lemma 4.1
  • ...and 2 more