Stefan problem

Problem

The one-dimensional one-phase Stefan problem describes a solid at its melting temperature $0$ that occupies the half-line $x > 0$ and melts when heat enters at $x = 0$. Let $a > 0$ be the thermal diffusivity of the liquid, $k > 0$ its thermal conductivity and $L > 0$ the latent heat per unit volume. Let $g \colon [0, \infty) \to [0, \infty)$ be a given boundary temperature. Find a function $s \colon [0, \infty) \to [0, \infty)$ with $s(0) = 0$, the position of the interface, and a temperature $u(x, t)$ defined for $0 \le x \le s(t)$, such that:

1. $\partial_t u = a \, \partial_x^2 u$ for $0 < x < s(t)$ and $t > 0$;
2. $u(0, t) = g(t)$ for $t > 0$;
3. $u(s(t), t) = 0$ for $t > 0$;
4. $L \, s'(t) = -k \, \partial_x u(s(t), t)$ for $t > 0$.

The unknown interface $s$ is the free boundary. Condition 4, the Stefan condition, says that the heat flux arriving at the interface is used to melt the solid.