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

Matthieu Alfaro (1) , Hiroshi Matano (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 GSMS - Graduate School of Mathematical Sciences

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.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - Series B, 2012, 17 (6), pp.1639-1649. ⟨10.3934/dcdsb.2012.17.1639⟩. ⟨hal-00641266⟩