On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions

• Published in: Nonlinear analysis Vol. 113; pp. 190 - 208
• Main Authors: Cârjă, Ovidiu, Miranville, Alain, Moroşanu, Costică
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.01.2015
• Subjects: Boundary conditions; Dynamic boundary conditions; Dynamical systems; Leray–Schauder principle; Mathematical analysis; Mathematical models; Nemytskii’s operator; Nonlinear dynamics; Nonlinear initial–boundary value problems; Nonlinear parabolic systems; Nonlinearity; Phase-field models; Regularity; Thermodynamics; Uniqueness; Nonlinear initial–boundary value problems; 80A22; 35K61; Nemytskii’s operator; Dynamic boundary conditions; Leray–Schauder principle; Phase-field models; Thermodynamics; 47H30; Nonlinear parabolic systems; 35Q56; 74A15; 35B65
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2014.10.003

This paper studies a Caginalp phase-field transition system endowed with a general regular potential, as well as a general class, in both unknown functions, of nonlinear and non-homogeneous (depending on time and space variables) boundary conditions. We first prove the existence, uniqueness and regularity of solutions to the Allen–Cahn equation, subject to the nonlinear and non-homogeneous dynamic boundary conditions. The existence, uniqueness and regularity of solutions to the Caginalp system in this new formulation are also proved. This extends previous works concerned with regular potential and nonlinear boundary conditions, allowing the present mathematical model to better approximate the real physical phenomena, especially phase transitions.