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

TL;DR

This paper investigates whether formal asymptotic expansions accurately describe solutions to the Allen-Cahn and FitzHugh-Nagumo equations with general initial data, confirming their validity for a broad class of solutions.

Contribution

It proves the validity of the principal term of formal asymptotic expansions for solutions with generic initial data in Allen-Cahn and FitzHugh-Nagumo systems.

Findings

Validation of formal expansions for a large class of solutions

Extension of previous results on asymptotic profiles

Use of eternal solutions to establish accuracy

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.