A multilayer mean-field model of a solid-liquid interface (SLI) is studied. The interface stability diagram is constructed on the basis of both discrete and the continuous approach taking into account the multistability phenomenon. An analytical expression is derived for the demarcation line, which separates two different SLI propagation modes: a barrier-controlled growth and an activationless one. The dynamics of the SLI propagation at high supercooling conditions is considered, the SLI runaway phenomenon accompanied by the SLI kinetical roughening is predicted.