  In this paper, we consider a system of initial boundary value problems for
parabolic equations, as a generalized version of the "$ \phi $-$ \eta $-$
\theta $ model" of grain boundary motion, proposed by Kobayashi [16]. The
system is a coupled system of: an Allen--Cahn type equation as in (1.1) with a
given temperature source; and a phase-field model of grain boundary motion,
known as "Kobayashi--Warren--Carter type model". The focus of the study is on a
special kind of solution, called energy-dissipative solution, which is to
reproduce the energy-dissipation of the governing energy in time. Under
suitable assumptions, two Main Theorems, concerned with: the existence of
energy-dissipative solution; and the large-time behavior; will be demonstrated
as the results of this paper.
