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 = A u + 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 u t = u x x + f ( x ) h ( t ) , from the final overdetermination or from a single point time measurement are presented.