An inverse problem and a time-like Carleman estimate for parabolic integro-differential equations

TL;DR

This paper addresses an inverse problem for parabolic integro-differential equations, establishing uniqueness and stability of source identification using a novel 'time-like' Carleman estimate and the Bukhgeim-Klibanov method.

Contribution

It introduces a new 'time-like' Carleman estimate for parabolic integro-differential equations and applies it to prove uniqueness and stability in inverse source problems.

Findings

Proved uniqueness of the inverse source problem.

Established a Hölder stability estimate.

Derived a new 'time-like' Carleman estimate.

Abstract

In this article, We investigate an inverse problem of determining the time-dependent source factor in parabolic integro-differential equations from boundary data. We establish the uniqueness and the conditional stability estimate of H\"older type for the inverse source problem in a cylindrical shaped domain. Our methodology is based on the Bukhgeim-Klibanov method by means of the Carleman estimate. Here we also derive the "time-like" Carleman estimate for parabolic integro-differential equations.