A note on uniqueness in the identification of a spacewise dependent source and diffusion coefficient for the heat equation
It provides theoretical uniqueness guarantees for an inverse problem of interest to applied mathematicians and engineers, but the result is incremental as it extends existing Carleman-based techniques to a specific combined inversion.
This paper proves uniqueness for simultaneously recovering a spacewise-dependent diffusion coefficient and heat source in the heat equation from a single final-time temperature measurement, using Carleman estimates in multi-dimensions and integral representations in 1D.
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 the derivatives of the conductivity, the source and the 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 of 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. Keywords: uniqueness; spacewise conductivity and source; final time measurements; heat equation; Carleman estimates.