TY - JOUR
T1 - Coarsening fronts
AU - Scheel, Arnd
PY - 2006/8/1
Y1 - 2006/8/1
N2 - We characterize the spatial spreading of the coarsening process in the Allen-Cahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the Allen-Cahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and the Conley index for ill-posed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy which depends on the existence of pulled fronts. The main tools here are an Evans function type construction for the infinite-dimensional ill-posed dynamics and an analysis of the complex dispersion relation based on Sturm-Liouville theory.
AB - We characterize the spatial spreading of the coarsening process in the Allen-Cahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the Allen-Cahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and the Conley index for ill-posed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy which depends on the existence of pulled fronts. The main tools here are an Evans function type construction for the infinite-dimensional ill-posed dynamics and an analysis of the complex dispersion relation based on Sturm-Liouville theory.
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U2 - 10.1007/s00205-006-0422-9
DO - 10.1007/s00205-006-0422-9
M3 - Article
AN - SCOPUS:33745587781
SN - 0003-9527
VL - 181
SP - 505
EP - 534
JO - Archive For Rational Mechanics And Analysis
JF - Archive For Rational Mechanics And Analysis
IS - 3
ER -