# On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

## TL;DR

This paper proves the uniqueness of identifying nonlinear diffusion coefficients in certain parabolic and elliptic equations using partial boundary data, without needing detailed knowledge of lower order terms.

## Contribution

It establishes a general uniqueness result for inverse problems of nonlinear diffusion coefficients, employing boundary data and singular test functions without assumptions on lower order terms.

## Key findings

- Uniqueness of nonlinear diffusion coefficient identification proven
- Method applicable with partial boundary data
- No need for detailed lower order term information

## Abstract

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail.