Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach ✩

The problem of determining the pair w := {F(x, t);T0(t)} of source terms in the parabolic equation ut = (k(x)ux )x + F(x, t) and Robin boundary condition −k(l)ux (l, t) = v[u(l, t) − T0(t)] from the measured final data μT (x) = u(x, T ) is formulated. It is proved that both components of the Fréchet gradient of the cost functional J(w)= μT (x)−u(x, T : w) 20 can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method. © 2006 Elsevier Inc. All rights reserved.