Uniqueness of Inverse Source Problems for General Evolution Equations

TL;DR

This paper establishes the uniqueness of inverse source problems for various evolution equations using interior observations, allowing flexible observation times and without initial non-vanishing assumptions, advancing the theoretical understanding of source identification.

Contribution

It introduces novel uniqueness results for inverse source problems in evolution equations, accommodating flexible observation times and separated source components without initial non-vanishing conditions.

Findings

Proved uniqueness for inverse source problems with separated variables.

Allowed flexible observation times not necessarily starting at t=0.

Combined Titchmarsh convolution theorem with unique continuation and time-analyticity.

Abstract

In this article, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at $t=0$, which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from $t=0$, which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.