The Calder\'on problem for a nonlocal diffusion equation with time-dependent coefficients

TL;DR

This paper establishes the global uniqueness for an inverse problem involving a nonlocal diffusion equation with time-dependent coefficients, extending previous elliptic results to parabolic cases across all spatial dimensions.

Contribution

It introduces a novel analysis of nonlocal Neumann derivatives and proves a global uniqueness theorem for the Calderón problem with time-dependent coefficients.

Findings

Partial exterior Dirichlet-to-Neumann map determines coefficients locally.

Interior determination is achieved through new analysis of nonlocal Neumann derivatives.

Global uniqueness holds for all spatial dimensions n ≥ 1.

Abstract

We investigate global uniqueness for an inverse problem for a nonlocal diffusion equation on domains that are bounded in one direction. The coefficients are assumed to be unknown and isotropic on the entire space. We first show that the partial exterior Dirichlet-to-Neumann map locally determines the diffusion coefficient in the exterior domain. In addition, we introduce a novel analysis of nonlocal Neumann derivatives to prove an interior determination result. Interior and exterior determination yield the desired global uniqueness theorem for the Calder\'on problem of nonlocal diffusion equations with time-dependent coefficients. This work extends recent studies from nonlocal elliptic equations with global coefficients to their parabolic counterparts. The results hold for any spatial dimension $n\geq 1$.