  In this paper, a class of systems of pseudo-parabolic PDEs is considered.
These systems (S)$_\varepsilon$ are derived as a pseudo-parabolic dissipation
system of Kobayashi--Warren--Carter energy, proposed by [Kobayashi et al.,
Physica D, 140, 141--150 (2000)], to describe planar grain boundary motion.
These systems have been studied in [arXiv:2402.10413], and solvability,
uniqueness and strong regularity of the solution have been reported under the
setting that the initial data is sufficiently smooth. Meanwhile, in this paper,
we impose weaker regularity on the initial data, and work on the weak
formulation of the systems. In this light, we set our goal of this paper to
prove two Main Theorems, concerned with: the existence and the uniqueness of
weak solution to (S)$_\varepsilon$, and the continuous dependence with respect
to the index $\varepsilon$, initial data and forcings.
