The diffusion equation with a jump change in diffusion coefficient depending on the diffusant concentration is considered. The phase transition problem with moving boundary to describe the features of activated recrystallization is formulated. An analytical description of the motion of the activated recrystallization front in the presence of a thin coating, which causes changes in the microstructure and physical properties of polycrystalline metals, is derived. The nonlinear equation, the solution of which describes the motion of activated recrystallization front, is found. It is shown that the dependence of the depth of the recrystallized layer is determined by such structural factors as the average size of recrystallized grains, the fraction of stationary grain boundaries, and jump in the average concentration of impurities in the zone of the front of activated recrystallization. The physical interpretation of coefficient of the Stefan condition at the moving boundary is given.