A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. (July 2017) - Colli, Pierluigi; Gilardi, Gianni; Nakayashiki, Ryota; Shirakawa, Ken
Abstract: 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 L 2 -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.