Free boundary problem governed by a non-linear diffusion-convection equation with Neumann condition

We consider a one-dimensional free boundary problem governed by a nonlinear diffusion - convection equation with a Neumann condition at fixed face $x=0$, which is variable in time and a like Stefan convective condition on the free boundary. Through successive transformations, an integral representation of the problem is obtained that involves a system of coupled nonlinear integral equations. Existence of the solution is obtained for all times by using fixed point theorems.