  In this paper, we consider a coupled system, known as
Kobayashi--Warren--Carter system, abbreviated as the KWC system. KWC system
consists of an Allen--Cahn type equation and a singular diffusion equation, and
it was proposed by [Kobayashi et al, Phys. D, 140, 141--150 (2000)] as a
possible mathematical model of grain boundary motion. The focus of this work is
on the dynamic boundary condition imposed in our KWC system, and the
mathematical interest is in a conflicting situation between: the continuity of
the transmission condition included in the dynamic boundary condition; and the
discontinuity encouraged by the singular diffusion equation. On this basis, we
will prove the Main Theorem concerned with the existence of solution to our KWC
system with energy-dissipation. Additionally, as a sub-result, we will prove a
key-lemma that is to give a certain mathematical interpretation for the
conflicting situation.
