Existence of an exact solution for a one-phase Stefan problem with nonlinear thermal coefficients from Tirskii’s method

The mathematical analysis of a one-phase Lamé–Clapeyron–Stefan problem with nonlinear thermal coefficients following [G.A. Tirskii, Two exact solutions of Stefan’s nonlinear problem, Sov. Phys. Dokl. 4 (1959) 288–292] is obtained. Two related cases are considered; one of them has a temperature condition on the fixed face x = 0 and the other one has a flux condition of the type − q 0 / t ( q 0 > 0 ) . We obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point.