An inequality for the coefficient $\sigma$ of the free boundary $s(t)=2\sigma \sqrt {t}$ of the Neumann solution for the two-phase Stefan problem

We consider a semi-infinite body (e.g. ice), represented by $\left ( {0, + \infty } \right )$, with an initial temperature $- c < 0$ having a heat flux $h\left ( t \right ) = - {h_0}/\sqrt t \left ( {{h_0} > 0} \right )$ in the fixed face $x = 0$. If ${h_0} > c{k_1}/\sqrt {\pi {a_1}}$ there exists a solution, of Neumann type, for the resulting two-phase Stefan problem. If we connect it with the Neumann problem (on $x = 0$ the body has a temperature $b > 0$ we obtain the inequality erf$\left ( {\sigma /{a_2}} \right ) < \left ( {{k_2}b{a_1}/{k_1}c{a_2}} \right )$ for the coefficient $\sigma$ of the free boundary $s\left ( t \right ) = 2\sigma \sqrt t$, where ${k_i}$, and $a_i^2$ are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding $i$ phase $i = 1:$ solid phase, $i = 2:$ liquid phase). If ${h_0} < c{k_1}/\sqrt {\pi {a_1}}$ there is no solution of the initial problem and if ${h_0} = c{k_1}/\sqrt {\pi {a_1}}$ the problem has no physical meaning and corresponds to the case where the latent heat of fusion $L$ tends to infinity.