Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

TL;DR

This paper addresses the inverse problem of reconstructing a degenerate diffusion coefficient in a parabolic equation, establishing uniqueness, stability, and numerical verification of the reconstruction methods.

Contribution

It introduces new theoretical results on uniqueness and stability for identifying degenerate diffusion coefficients and powers, using energy methods and Carleman estimates.

Findings

Proved Lipschitz stability for constant coefficients and powers.

Established uniqueness for general diffusion coefficients and powers.

Numerically verified the theoretical reconstruction methods.

Abstract

We consider the inverse problem of identification of degenerate diffusion coefficient of the form $x^\alpha a(x)$ in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient $a$ and the power $\alpha$ by knowing an interior data at some time. On the other hand, we obtain the uniqueness result for the identification of a general diffusion coefficients $a(x)$ and also the power $\alpha$ form a boundary data on one side of the space interval. The proof is based on global Carleman estimates for a hyperbolic problem and an inversion of the integral transform similar to the Laplace transform. Finally, the theoretical results are satisfactory verified by numerically experiments.