The Calder\'{o}n problem for the fractional wave equation: Uniqueness and optimal stability

TL;DR

This paper investigates the inverse problem for the fractional wave equation, establishing uniqueness and optimal stability estimates for determining the potential from exterior measurements, highlighting differences from classical wave equations.

Contribution

It proves uniqueness and optimal log-type stability estimates for the fractional wave equation inverse problem, using unique continuation properties of the fractional Laplacian.

Findings

Uniqueness of potential determination from exterior data.

Optimal log-type stability estimate established.

Differences between fractional and classical wave equations in stability.

Abstract

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension $n\in \N$.