Lipschitz stability in inverse source problems for a singular parabolic equation

ABSTRACT

In this paper, we study inverse source problems for the heat equation with an inverse-square potential localized on the boundary of a smooth bounded domain, and with a locally distributed observation. We first derive an improved Carleman estimate of that obtained by Cazacu [SIAM J Control Optim. 2014;52:2055–2089] and then use it to prove the Lipschitz stability result for the inverse source problem.

COMMUNICATED BY:

• M. Yamamoto

Keywords:

• Carleman estimate
• Hardy inequality
• inverse source problem
• Lipschitz stability
• parabolic equation
• inverse-square potential

2010 Mathematics Subject Classifications:

• 34M50
• 35K67
• 35A23

Acknowledgements

The authors would like to thank the reviewers for their helpful comments and suggestions which improved the presentation of the paper. This work was completed while the first and second authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank the Institute for its hospitality and support.

Disclosure statement

No potential conflict of interest was reported by the author(s).