Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation

Corrado Lattanzio (1) , Corrado Mascia (2) , Ramon G. Plaza (3) , Chiara Simeoni (4)

1 DISIM - Dipartimento di Ingegneria, Scienza dell'Informazione e Matematica
2 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
3 IIMAS - UNAM - Departamento de Matemática y Mecánica
4 PDEs and Numerical Analysis

A modification of the parabolic Allen-Cahn equation, determined by the substitution of Fick's diffusion law with a relaxation relation of Cattaneo-Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

Mots clés: Allen-Cahn equation; Traveling waves; Nonlinear stability

Domaines: Equations aux dérivées partielles [math.AP]

Corrado Lattanzio, Corrado Mascia, Ramon G. Plaza, Chiara Simeoni. Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation. 2014. ⟨hal-01072141⟩