Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Determining a fractional Helmholtz equation with unknown source and scattering potential

Pages: 1861 – 1876



Xinlin Cao (Department of Mathematics, Hong Kong Baptist University, Kowloon, H.K.)

Hongyu Liu (Department of Mathematics, City University of Hong Kong, Kowloon, H.K.)


We are concerned with an inverse problem associated with a fractional Helmholtz equation that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequency-dependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.


fractional Helmholtz equation, simultaneous recovery, low-frequency asymptotics, compact embedding theorem, strong uniqueness property, Runge approximation property

2010 Mathematics Subject Classification

26A33, 35J05, 35P25, 35R30

Received 30 July 2018

Accepted 3 June 2019

Published 6 January 2020