# Uniqueness to some inverse source problems for the wave equation in unbounded domains

## TL;DR

This paper establishes uniqueness results for inverse source problems in unbounded domains for the wave equation, demonstrating how boundary data can recover specific source terms, moving sources, and embedded obstacles using Laplace transform techniques.

## Contribution

It provides new uniqueness theorems for inverse acoustic source problems with various source forms and moving sources, extending previous results in unbounded domains.

## Key findings

- Unique recovery of source terms of specific forms from boundary data.
- Ability to recover moving point sources and their orbits with minimal data.
- Simultaneous identification of obstacles and sources in inhomogeneous media.

## 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 proven 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.