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.  2005 Elsevier Inc. All rights reserved