Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Abstract

This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f (x)g(t) and f (x 1 , x 2 , t)h(x 3 ), where g(t) and h(x 3 ) are given and x = (x 1 , x 2 , x 3 ) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.problem for doubly hyperbolic equations arising from the nucleation rate reconstruction in the three-dimensional time cone model was analyzed in [32]. A Lipschitz stability result was proved for recovering the spatial component of the source term using interior data and an iterative thresholding algorithm (see also [26] with the final observation data) was tested. However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated. The increasing stability analysis in recovering spatial-dependent source terms has been carried out from both theoretical and numerical points of view (see e.g., [3,6,4,5,7,11,31,40]).In the time domain, it is very natural to transform the wave scattering problem governed by hyperbolic equations into elliptic inverse problems in the Fourier or Laplace domain with multi-frequency data; see e.g. [24] for determining sound-hard and impedance obstacles in a homogeneous background medium. In [7], the time-domain analysis helps for deriving an increasing stability to time-harmonic inverse source problems via Fourier transform. The same idea was used in [2,20,19] for recovering spatial-dependent sources as well as moving source profiles and orbits in elastodynamics and electromagnetism. The aim of this paper is to analyze the acoustic counterpart with new uniqueness results. Specially, this paper concerns the following four inverse problems with a single boundary surface data:1. Simultaneous determination of sound-soft obstacles and separable source terms in an inhomogeneous medium (Subsection 2.1).2. Simultaneous determination of sound-soft obstacles and general time-dependent source terms from a family of solutions (Subsection 2.2).3. Inverse moving point source pro...

Inverse moving source problem for time-fractional evolution equations: Determination of profiles

“…Remark 3.1 The proof of Lemma 2.3 for α ∈ (1, 2) relies heavily on the analyticity of the solution in the time variable, which applies to the scalar wave equation (α = 2) when the dynamical measurement data over (0, ∞) are available; see [23,Theorem 2.1 and Corollary 2.3] where the data are measured on a closed surface. In Subsection 4.3 below, we shall present a proof for the wave equation using the data over a finite time period (0, T ).…”

Determination of source and initial values for acoustic equations with a time-fractional attenuation

“…The above-cited articles are concerned with hyperbolic inverse source problems in a bounded spatial domain. Inverse source problems in an unbounded domain were studied in [3,10,11] and in [12], where the source term and an obstacle were retrieved. As for the recovery of initial data in the background of the TAT or PAT, we refer the reader to [2,9,29].…”

“…where α 2 := 1+α1 2 and C 2 is a positive constant depending only on Ω, T , α and q. Indeed, since α and q lie in W 1,∞ (Ω), then it is clear from (12) that Q(t) ∈ B(H) for all t ∈ (0, T ]. Moreover, we have…”

The inverse source problem for the wave equation revisited: A new approach

“…Under a weak regularity assumption, a uniqueness result for a multidimensional hyperbolic inverse source problem with a single measurement was proved in [28]. Recently, an approach of using Laplace transform was employed in [12] to show the uniqueness of some inverse source problems for the wave equation.…”