Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term

We study the nonhomogeneous heat equation under the form ut−uxx=φ(t)f(x), where the unknown is the pair of functions (u,f). Under various assumptions about the function φ and the final value in t=1, i.e., g(x), we propose different regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For φ≢0 the solution is unique.