I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary does not have the nice "solution $u=0$ " criteria.

Instead, it has a robin boundary condition
$$u+\partial_{\nu}u = g,$$
where $g$ is a given function and $\partial_{\nu}$ denotes the derivative in direction of the normal vector to the free boundary, at *the free boundary* (not fixed boundary).

(The above robin condition at the free boundary is assigned in addition to the typical condition on the velocity of the free boundary: $v = \partial_{\nu} u^+ - \partial_{\nu}u^-$.)

Does anyone know if this is an open problem? Or does anyone know a work related to this? I will greatly appreciate any reference to a 2-phase Stefan-type problem that is not the classical Stefan problem. Thank you!