On the validity of formal asymptoticexpansions in Allen-Cahn equation and FitzHugh-Nagumo system withgeneric initial data

Matthieu Alfaro; Hiroshi Matano

doi:10.3934/dcdsb.2012.17.1639

Article Contents

2012, Volume 17, Issue 6: 1639-1649. Doi: 10.3934/dcdsb.2012.17.1639

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

Matthieu Alfaro^1, and
Hiroshi Matano^2,

1.
Univ. Montpellier 2, I3M, UMR CNRS 5149, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5
2.
Graduate school of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914

Received: November 2011

Revised: February 2012

Published: September 2012

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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.

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Mathematics Subject Classification: Primary: 35B25, 35C20, 35R35.

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References

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