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

Abstract

This paper considers the weakly coupled parabolic system ${\partial }_{t}u-{\partial }_x^{2}u+P(x)u=0$ 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.