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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 ina bounded spacial domain $ \Omega $, and another Allen-Cahn type equation onthe smooth boundary $ \Gamma := \partial \Omega $, and besides, these coupledequations are transmitted via the dynamic boundary conditions. In particular,the equation in $ \Omega $ is derived from the non-smooth energy proposed byVisintin in his monography "Models of phase transitions": hence, the diffusionin $ \Omega $ is provided by a quasilinear form with singularity. The objectiveof 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 $. On this basis, wewill prove two Main Theorems 1 and 2, which will be concerned with thewell-posedness of (ACE)$_\varepsilon$ for each $ \varepsilon \geq 0 $, and thecontinuous dependence of solutions to (ACE)$_\varepsilon$ for the variations of$ \varepsilon \geq 0 $, respectively.

A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions