Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

TL;DR

This paper establishes a new logarithmic stability estimate for identifying a time-dependent zero order coefficient in a parabolic PDE from partial boundary data, without requiring measurements at the final time, and applies it to nonlinear term determination.

Contribution

It introduces a novel stability estimate avoiding final time measurements and extends it to nonlinear term identification in semilinear parabolic equations.

Findings

New stability estimate for zero order coefficient without final time data

Application to nonlinear term determination in semilinear equations

Use of a parabolic Carleman inequality for constructing solutions

Abstract

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann. This parabolic Carleman inequality enters in an essential way in the construction of complex geometric optics solutions that vanish at a part of the lateral boundary.