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We investigate uniqueness in the inverse problem of reconstructingsimultaneously a spacewise conductivity function and a heat source in theparabolic heat equation from the usual conditions of the direct problem andadditional information from a supplementary temperature measurement at a givensingle instant of time. In the multi-dimensional case, we use Carlemanestimates for parabolic equations to obtain a uniqueness result. The given dataand the solution domain are sufficiently smooth such that the required normsand the derivatives of the conductivity, the source and the solution of theparabolic heat equation exist and are continuous throughout the solutiondomain. These assumptions can be further relaxed using more involved estimatesand techniques but these lengthy details are not included. Instead, in thespecial case of the one-dimensional heat equation, we give an alternative andrather straightforward proof of uniqueness of the inverse problem, based onintegral representations of the solution together with density results forsolutions of the corresponding adjoint problem. In this case, the requiredregularity conditions on the conductivity, source and the solution of theparabolic heat equation are weakened to classes of integrable functions.  Keywords: uniqueness; spacewise conductivity and source; final timemeasurements; heat equation; Carleman estimates.

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