Recovering a scalar time dependent function in a multidimensional parabolic equation by a nonlocal boundary additional information

In this paper we solve (locally in time and under suitable assumptions on the data) an identification problem related to a linear parabolic equation when the additional information is time-dependent and nonlocal in space. More exactly, our problem consists in recovering a (positive) time-dependent coefficient β in front of the time derivative. We prove a local in time existence, uniqueness and stability result when the data belong to suitable function spaces. Our basic tool is the Semigroup Theory of linear bounded operators.