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

Hiroshi Matano,Matthieu Alfaro

doi:10.3934/dcdsb.2012.17.1639

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.

Highlights

In this paper, we study the behavior of solution uε of an Allen-Cahn type equation of the form ut = ∆u + 1 ε2 (f (u) − εgε(x, t, u)) in Ω × (0, ∞) (Pε) ∂u
Our goal is to provide an affirmative answer in this direction: we shall prove that, for ε sufficiently small, the solution uε — with rather general initial data — of both (Pε) and (RDε) possesses a profile that agrees with the principal term of the formal expansion
Let us emphasize that we do not assume that the initial data u0 of uε already has well-developed transition layers depending on ε, in which case the validity of the formal expansions is more or less known

Summary

Introduction

We study the behavior of solution uε of an Allen-Cahn type equation of the form. Let us emphasize that we do not assume that the initial data u0 of uε already has well-developed transition layers depending on ε, in which case the validity of the formal expansions is more or less known (see subsection 1.2 for more details). In [2], the present authors improved this estimate to O(ε) They show that the solution uε develops a steep transition layer within the time scale of O(ε2| ln ε|), and that the layer obeys the law of motion that coincides with the formal asymptotic limit (P0) or (RD0) within an error margin of O(ε). — as far as the Allen-Cahn equation is concerned — following [17], [15] one can define a limit problem for all t ≥ 0 that generalizes (P0) in the framework of viscosity solutions. In this setting we refer to [16] (convergence of Allen-Cahn equation with prepared initial data to generalized motion by mean curvature), [4], [6] (generalizations), [22, 23], [7], [5] (not well-prepared initial data), [1] (fine convergence rate)

Main results

Proof of the main results