Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain

Abstract

We consider the Fife-Greenlee problemε2Δu+(u−a(y))(1−u2)=0in Ω,∂u∂ν=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, ε>0 is a small parameter, ν denotes the unit outward normal of ∂Ω. Let Γ={y∈Ω:a(y)=0} be a simple smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two disjoint nonempty components. We assume that −1<a(y)<1 on Ω and ∇a≠0 on Γ, and also some admissibility conditions hold for a, Γ and ∂Ω. For any fixed integer N=2m+1≥3, we will show the existence of a clustered solution uε with N-transition layers near Γ with mutual distance O(ε|log⁡ε|), provided that ε stays away from a discrete set of values at which resonance occurs.