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 R m + n . The supplementary information, which is necessary to solve it, is given by the knowledge of ∫ R n u ( t , x , y ) d μ ( y ) for every ( t , x ) , where u is the solution of the parabolic problem, and μ is a complex valued Borel measure in R n .