Partial reconstruction of the source term in a linear parabolic initial value problem

We consider a problem of partial reconstruction of the source term, together with the solution, in a linear parabolic Cauchy problem in Rm+n. The supplementary information, which is necessary to solve it, is given by the knowledge of for every (t,x), where u is the solution of the parabolic problem, and μ is a complex valued Borel measure in Rn.