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)ε for ε≥0. For each ε≥0, the system (ACE)ε consists of an Allen–Cahn type equation in a bounded spacial domain Ω, and another Allen–Cahn type equation on the smooth boundary Γ:=∂Ω, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the non-smooth energy proposed by Visintin in his monography “Models of phase transitions”: hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L2-based solutions to our systems, and to see some robustness of (ACE)ε with respect to ε≥0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE)ε for each ε≥0, and the continuous dependence of solutions to (ACE)ε for the variations of ε≥0, respectively.