A one-phase Stefan problem for a non-classical heat equation with a heat flux condition on the fixed face

We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a heat flux boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0. We use the Friedman–Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.