Partial reconstruction of the source term in a linear parabolic problem

We consider, in some different situations, the problem of the reconstruction of the source term in a parabolic problem in a space-time domain [0, T] × I × Ω: this source term is assumed of the form g(t,x) f(t,x,y) (t ∈ [0, T], x ∈ I, y ∈ Ω), with f given and g to be determined. The novelty, with respect to the existing literature, lies in the fact that g depends on time and on some of the space variables. The supplementary information, allowing to determine g together with the solution of the problem u, is given by the knowledge, for every (t,x), of an integral of the form ∫{Ω} u(t,x,y) dμ(y), with μ complex Borel measure.