A phase transition model for the helium supercooling

We build a predictive theory for the evolution of mixture of helium and supercooled helium at low temperature. The absolute temperature and the volume fraction of helium, which is dominant at temperature larger than the phase change temperature, are the state quantities. The predictive theory accounts for local interactions at the microscopic level, involving the gradient of $\beta$. The nonlinear heat flux in the supercooled phase results from a Norton-Hoff potential. We prove that the resulting set of partial differential equations has solutions within a convenient analytical frame.