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)?? for ?>=0?>=0. For each ?>=0?>=0, the system (ACE)?? consists of an Allen-Cahn type equation in a bounded spacial domain ??, and another Allen-Cahn type equation on the smooth boundary ?:=???:=??, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in ?? is derived from the non-smooth energy proposed by Visintin in his monography "Models of phase transitions": hence, the diffusion in ?? is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L2L2-based solutions to our systems, and to see some robustness of (ACE)?? with respect to ?>=0?>=0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE)?? for each ?>=0?>=0, and the continuous dependence of solutions to (ACE)?? for the variations of ?>=0?>=0, respectively.