Many inverse heat conduction problems lead us to consider the following noncharacteristic Cauchy problem for parabolic equations of the form“surface temperature”“surface heat flux”wherepis an elliptic operator, ϕ andgare given functions. This problem is well-known to be severely ill-posed, and up to now there have been many approaches for solving it in a stable way. However, most of them need a supplementary condition: either the initial condition, or a boundary condition, etc ⃜In this paper a variational method for this problem is suggested. In contrast to the other works, in the paper the initial condition is not assumed to be known. A short discussion on using the gradient methods is also given.