The global-in-time existence of a classical solution for some free boundary problem

We consider the problem with free (unknown) boundary for the one-dimensional diffusion-convection equation. The unknown boundary is found from the additional condition on the free boundary. A dilation of the variables reduces the problem to an initial-boundary value problem for a strictly parabolic equation with unknown coefficients in the known domain. These coefficients are found from an additional boundary condition, which makes it possible to construct a nonlinear operator whose fixed points define the solution to the initial problem.