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In this paper, we consider a system of initial‐boundary value problems for parabolic equations, as a generalized version of the “ φ ‐ η ‐ θ  model” of grain boundary motion, proposed by Kobayashi (2001). The system is a coupled system of an Allen–Cahn‐type equation 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 and the large‐time behavior, will be demonstrated as the results of this paper.

Energy‐dissipation in a coupled system of Allen–Cahn‐type equation and Kobayashi–Warren–Carter‐type model of grain boundary motion