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 ut=Au+Fut=Au+F, when the measured output data is given in the form the final overdetermination uT(x)≔u(x,T)uT(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)f(x) and h(t)h(t) of the heat equation ut=uxx+f(x)h(t)ut=uxx+f(x)h(t), from the final overdetermination or from a single point time measurement are presented.