Local logarithmic stability of an inverse coefficient problem for a singular heat equation with an inverse-square potential

Abstract

An inverse problem of the determination of the coefficient $P(x)$ in the equation: ${\mathrm{\partial }}_{t}u(x,t)-\mathrm{\nabla }\cdot (P(x)\mathrm{\nabla }u)-\frac{\mu }{|x{|}^2}u(x,t)=0,(x,t)\in \mathrm{\Omega }\times (0,T)$ is considered. The main difficulty here as compared with the existing result is that there is a singular potential in the equation. A local logarithmic stability estimate is obtained using the method of Carleman estimates. Our proof relies on the Bukhgeim–Klibanov method which was originated in [Bukhgeim AL, Klibanov MV. Uniqueness in the large of a class of multidiimensional inverse problems. Dokl Akad Nauk SSSR. 1981;17:244–247] to prove inverse source or coefficient results.

Keywords:

• Singular heat equation
• logarithmic stability
• inverse coefficient problem

MSC:

• 35B37
• 93B05
• 35K05
• 49J20

Acknowledgments

The authors thank anonymous reviewers for the valuable comments.

Disclosure statement

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

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11971119].