On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data

Pages: 1639 - 1649, Volume 17, Issue 6, September 2012

doi:10.3934/dcdsb.2012.17.1639       Abstract        References        Full Text (365.1K)       Related Articles

Matthieu Alfaro - Univ. Montpellier 2, I3M, UMR CNRS 5149, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (email)
Hiroshi Matano - Graduate school of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914, Japan (email)

Abstract: Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

Keywords:  Singular perturbation, asymptotic expansion, front profile, reaction-diffusion system.
Mathematics Subject Classification:  Primary: 35B25, 35C20, 35R35.

Received: November 2011;      Revised: February 2012;      Published: May 2012.

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