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

Abstract

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.