  In this paper, we consider a class of coupled systems of PDEs, denoted by
(ACE)$_{\varepsilon}$ for $ \varepsilon \geq 0 $. For each $ \varepsilon \geq 0
$, the system (ACE)$_{\varepsilon}$ consists of an Allen-Cahn type equation in
a bounded spacial domain $ \Omega $, and another Allen-Cahn type equation on
the smooth boundary $ \Gamma := \partial \Omega $, and besides, these coupled
equations are transmitted via the dynamic boundary conditions. In particular,
the equation in $ \Omega $ is derived from the non-smooth energy proposed by
Visintin in his monography "Models of phase transitions": hence, the diffusion
in $ \Omega $ is provided by a quasilinear form with singularity. The objective
of this paper is to build a mathematical method to obtain meaningful $ L^2
$-based solutions to our systems, and to see some robustness of
(ACE)$_\varepsilon$ with respect to $ \varepsilon \geq 0 $. On this basis, we
will prove two Main Theorems 1 and 2, which will be concerned with the
well-posedness of (ACE)$_\varepsilon$ for each $ \varepsilon \geq 0 $, and the
continuous dependence of solutions to (ACE)$_\varepsilon$ for the variations of
$ \varepsilon \geq 0 $, respectively.
