The Calder\'{o}n problem for nonlocal parabolic operators

TL;DR

This paper addresses inverse problems for nonlocal parabolic operators, reducing them to local problems, and establishes uniqueness and non-uniqueness results, along with a new proof of unique continuation.

Contribution

It introduces a method to convert nonlocal inverse problems into local ones and provides new theoretical insights and proofs for the Calderón problem in this context.

Findings

Reduced nonlocal inverse problems to local inverse problems

Established uniqueness and non-uniqueness results for nonlocal parabolic Calderón problems

Derived a new equation and proved unique continuation property

Abstract

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calder\'on problems, respectively.