Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions

TL;DR

This paper establishes the unique determination of an inverse source in a one-dimensional time-fractional diffusion equation using minimal data, leveraging unique continuation properties for weak solutions.

Contribution

It provides the first sharp uniqueness result for an inverse x-source problem in a time-fractional diffusion equation with minimal lateral data.

Findings

Sharp uniqueness result for inverse source problem

Unique continuation holds for weak solutions

Minimal lateral Cauchy data suffices for uniqueness

Abstract

In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique continuation which holds for weak solutions.