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

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.