We present a transient mathematical model for the sublimation growth of silicon carbide (SiC) single crystals by the physical vapor transport (PVT) method. The model of the gas phase consists of balance equations for mass, momentum, and energy, as well as reaction-diffusion equations. Due to physical and chemical reactions, the gas phase is encompassed by free boundaries. Nonlinear heat transport equations are considered in the various solid components of the growth system. Discontinuous and nonlocal interface conditions are formulated to account for temperature steps between gas and solid as well as for diffuse-gray radiative heat transfer between cavity surfaces. An axisymmetric induction heating model is devised using a magnetic scalar potential. For a nonlinear evolution problem arising from the model, a finite volume scheme is stated, followed by a discrete existence and uniqueness result. We conclude by presenting and analyzing results of transient numerical experiments relevant to the physical growth process.