  Two main existence theorems are proved for two nonstandard systems of
parabolic initial-boundary value problems. The systems are based on the "$ \phi
$-$ \eta $-$ \theta $ model" proposed by Kobayashi [RIMS Kokyuroku, 1210
(2001), 68-77] as a phase-field model of planar grain boundary motion under
isothermal solidification. Although each of the systems has specific
characteristics and mathematical difficulties, the proofs of the main theorems
are based on the time discretization method by means of a common approximating
problem. As a consequence, we provide a uniform solution method for a wide
scope of parabolic systems associated with the $ \phi $-$ \eta $-$ \theta $
model.
