An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation

TL;DR

This paper addresses the inverse problem of identifying the radiative coefficient in a degenerate parabolic equation using final data, establishing uniqueness, stability, and convergence of solutions within an optimal control framework.

Contribution

It introduces a novel approach for inverse coefficient problems in degenerate parabolic equations, including uniqueness, stability, and convergence analysis.

Findings

Proved uniqueness of the inverse problem solution.

Established stability and convergence of the minimizer.

Extended results to more general degenerate parabolic equations.

Abstract

This work investigates an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in the paper belongs to the second order parabolic equations with non-negative characteristic form, namely that there exists degeneracy on the lateral boundaries of the domain. The uniqueness of the solution is obtained by the contraction mapping principle. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some \emph{a-priori} regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in the paper. The results obtained in the paper are interesting and useful, and can be extended to more general degenerate parabolic equations.