A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. (July 2017)

• Record Type: Journal Article
• Authors: Colli, Pierluigi; Gilardi, Gianni; Nakayashiki, Ryota; Shirakawa, Ken
• Is Part Of: Nonlinear analysis. Volume 158(2017)
• Page Start: 32
• Page End: 59
• Publication Date: 2017-07
• Subjects: 35K55 -- 35K59 -- 82C26; Quasi-linear Allen–Cahn equation -- Dynamic boundary conditions -- Non-smooth energy functional -- Initial–boundary value problem -- Well-posedness -- Continuous dependence; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.03.020
• Languages: English
• ISSNs: 0362-546X

Abstract

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.