Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5024627 | Nonlinear Analysis: Theory, Methods & Applications | 2017 | 28 Pages |
Abstract
In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE)ε for εâ¥0. For each εâ¥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 Î:=âΩ, 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 εâ¥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 εâ¥0, and the continuous dependence of solutions to (ACE)ε for the variations of εâ¥0, respectively.
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Authors
Pierluigi Colli, Gianni Gilardi, Ryota Nakayashiki, Ken Shirakawa,