Two parabolic inverse problems for an equation with unbounded zero-order coefficient

TL;DR

This paper establishes uniqueness and stability results for inverse problems involving unbounded zero-order coefficients in parabolic equations, including a new global unique continuation principle for Schrödinger equations with unbounded potentials.

Contribution

It proves the largest class of unbounded zero-order coefficients can be uniquely determined and provides a new logarithmic stability estimate for initial condition recovery.

Findings

Uniqueness of coefficient determination from boundary data.

Logarithmic stability for initial condition from interior measurement.

A new global quantitative unique continuation for Schrödinger equations with unbounded potential.

Abstract

This work is composed of two parts. We prove in the first part the uniqueness of the determination of the unbounded zero-order coefficient in a parabolic equation from boundary measurements. The novelty of our result is that it covers the largest class of unbounded zero-order coefficients. We establish in the second part a logarithmic stability inequality for the problem of determining the initial condition from a single interior measurement. As by-product, we obtain an observability inequality for a parabolic equation with unbounded zero-order coefficient. The proof of this observability inequality is based on a new global quantitative unique continuation for the Schr\"odinger equation with unbounded potential. For the sake of completeness, we provide in Appendix A a full proof of this result.