  One-dimensional free boundary problem for a nonlinear diffusion - convection
equation with a Dirichlet condition at fixed face $x=0$, variable in time, is
considered. Throught several transformations the problem is reduced to a free
boundary problem for a diffusion equation and the integral formulation is
obtained. By using fixed point theorems, the existence of at least a solution,
for small time, to a system of coupled nonlinear integral equations is
obtained.
