The stability of the radiation front in the scrape-off layer (SOL) of a tokamak is studied with a one-dimensional fluid code; the time-dependent transport equations are solved in the direction parallel to a magnetic field line. As the energy from the core into the SOL plasma is reduced, stable attached solutions change into stable detached solutions. Whenever such stable detached states are attained, the strong radiation front is in contact with or at a small distance from the divertor target. When the input energy is decreased further and total losses become larger than the input power, the radiation front starts to move towards the X-point, cooling the SOL plasma. In such a case, no solutions for the radiation front resting in the divertor channel are observed in our parameter space. This qualitatively corresponds to the results of tokamak divertor experiments which show the movement of the radiation front.