Stability Estimates for Some Parabolic Inverse Problems With the Final Overdetermination via a New Carleman Estimate

TL;DR

This paper establishes stability estimates and uniqueness results for inverse coefficient and source problems in linear parabolic equations with final time data, using a novel Carleman estimate to handle partial boundary conditions.

Contribution

The paper introduces a new Carleman estimate to derive stability and uniqueness for inverse parabolic problems with final time data and partial boundary conditions.

Findings

Holder and Lipschitz stability estimates derived

Uniqueness theorems established for inverse problems

Applicable to problems with partial boundary data

Abstract

This paper is about Holder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time {t=T}. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.