An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach

This paper presents a semigroup approach for inverse source problems for the abstract heat equation U-t = Au + F, when the measured output data is given in the form the final overdetermination u(T)(x) := u(x, T). A representation formula for a solution of the inverse source problem is proposed. This representation shows a non-uniqueness structure of the inverse problem solution, and also permits one to derive a sufficient condition for uniqueness. Some examples related to identifying the unknown spacewise and time-dependent heat sources f (x) and h(t) of the heat equation 14, = u(t) + f (x)h(t), from the final overdetermination or from a single point time measurement are presented. (C) 2012 Elsevier Ltd. All rights reserved.