Recovering the initial condition in the one-phase Stefan problem

<p style='text-indent:20px;'>We consider the problem of recovering initial condition in one-dimensional one-phase Stefan for heat equation from knowledge position melting point. We first recall some properties free boundary solution. Then we study uniqueness and stability inversion. The principal contribution paper is a new logarithmic type estimate that shows inversion may be severely ill-posed. proof based on integral equations representation techniques, unique continuation property parabolic solutions. also present few numerical examples operating with noisy synthetic data.</p>