Uniqueness for an inverse coefficient problem of a weakly coupled parabolic system

TL;DR

This paper proves the unique determination of a coefficient matrix in a weakly coupled parabolic system from boundary observations, extending classical inverse problem techniques to systems.

Contribution

It introduces a novel approach combining eigenfunction expansion and Gel'fand-Levitan theory for inverse coefficient problems in parabolic systems.

Findings

Coefficient matrix P(x) is uniquely determined by boundary data.

Eigenfunction expansion is used to analyze the inverse problem.

Extension of Gel'fand-Levitan theory to systems is developed.

Abstract

This paper considers the weakly coupled parabolic system $\partial_t u-\partial^2_xu +P(x)u=0$ with the homogeneous Neumann boundary condition, where \(P(x)\) is a \(2\times2\) symmetric real-valued function matrix. Under the assumption that the initial value \(a(x)\) is a generating element (i.e., it has a nonzero inner product with every eigenfunction), we prove that the coefficient matrix $ P(x)$ is uniquely determined by the boundary observation $u(0, t)$, $u(1, t)$, $0 < t < T$. The proof relies on the eigenfunction expansion of the solution to the initial-boundary value problem and an extension of the Gel'fand-Levitan theory to the parabolic system.