summarise our results as follows:

-   In the pressure range where the three thermal equilibrium phase can coexist under a constant pressure $(P_\mathrm{min}<P_\mathrm{c}<P_\mathrm{max})$, we find solutions that connects the CNM and the unstable phase. The solutions can be classified into four type solutions, C-U-o, C-U, C-c, and C-C solutions in order of the mass flux. The C-U-o and C-U solutions connect the CNM and the unstable phase with and without the oscillation of physical quantities in the low-density side. The C-c solution is truncated at the maximum temperature above which there are no steady solutions. Combination of a C-c solution and a shock front provides an infinite extent solution. The C-C solutions connect the CNM and CNM.

-   We have also found steady solutions in the pressure range $(P_\mathrm{c}>P_\mathrm{max})$ where only CNM can be in equilibrium. This pressure range is realized in the shock compressed region. In this parameter space $(P_\mathrm{c},j)$, most of solutions belong to the C-c solution.

-   We derived a critical mass flux $j_\mathrm{crit}$ above which the effect of advection becomes important compared with the heat conduction in energy transfer. For large mass flux, $|j|>|j_\mathrm{crit}|$, the thickness of the front is characterized by the cooling length instead 