On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.

► We study a non-linear free boundary problem for a diffusion–convective equation.
► We reduce it to another free boundary problem by using the Bäcklund transformation.
► We prove that it is equivalent to solve a system of Volterra integral equations.
► We use the Banach contraction theorem following the Friedman–Rubinstein's method.