A note on uniqueness in the identification of a spacewise dependent source and diffusion coefficient for the heat equation

We investigate uniqueness in the inverse problem of reconstructing simultaneously a spacewise conductivity function and a heat source in the parabolic heat equation from the usual conditions of the direct problem and additional information from a supplementary temperature measurement at a given single instant of time. In the multi-dimensional case, we use Carleman estimates for parabolic equations to obtain a uniqueness result. The given data and the solution domain are sufficiently smooth such that the required norms and derivatives of the conductivity, source and solution of the parabolic heat equation exist and are continuous throughout the solution domain. These assumptions can be further relaxed using more involved estimates and techniques but these lengthy details are not included. Instead, in the special case of the one-dimensional heat equation, we give an alternative and rather straightforward proof of uniqueness for the inverse problem, based on integral representations of the solution together with density results for solutions of the corresponding adjoint problem. In this case, the required regularity conditions on the conductivity, source and the solution of the parabolic heat equation are weakened to classes of integrable functions.