Recovery of nonlinear terms for reaction diffusion equations from boundary measurements

TL;DR

This paper introduces a new method for uniquely and stably recovering general semilinear terms in nonlinear parabolic equations from boundary measurements, advancing inverse problem solutions in this domain.

Contribution

It develops a novel criterion and employs second linearization to achieve the first known stable recovery of semilinear terms from boundary data.

Findings

Proves unique recovery of semilinear terms from boundary measurements.

Establishes stability estimates for the inverse problem.

Introduces a second linearization approach for nonlinear inverse problems.

Abstract

We consider the inverse problem of determining a general semilinear term appearing in nonlinear parabolic equations. For this purpose, we derive a new criterion that allows to prove global recovery of some general class of semilinear terms from lateral boundary measurements of solutions of the equation with initial condition fixed at zero. More precisely, we prove, for what seems to be the first time, the unique and stable recovery of general semilinear terms depending on time and space variables independently of the solution of the nonlinear equation from the knowledge of the parabolic Dirichlet-to-Neumann map associated with the solution of the equation with initial condition fixed at zero. Our approach is based on the second linearization of the inverse problem under consideration.