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

作者：Colli Pierluigi; Gilardi Gianni; Nakayashiki Ryota^*; Shirakawa Ken

来源：Nonlinear Analysis-Theory Methods & Applications, 2017, 158: 32-59.

DOI：10.1016/j.na.2017.03.020

• Quasi-linear Allen-Cahn equation
• Dynamic boundary conditions
• Non-smooth energy functional
• Initial boundary value problem
• Well-posedness
• Continuous dependence

In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE)(epsilon) for epsilon >= 0. For each epsilon >= 0, the system (ACE)(epsilon) consists of an Allen-Cahn type equation in a bounded spacial domain ohm, and another Allen-Cahn type equation on the smooth boundary Gamma := partial derivative ohm, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in ohm is derived from the non-smooth energy proposed by Visintin in his monography "Models of phase transitions": hence, the diffusion in ohm 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)epsilon with respect to epsilon >= 0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE)epsilon for each epsilon >= 0, and the continuous dependence of solutions to (ACE)epsilon for the variations of epsilon >= 0, respectively.

• 出版日期2017-7