Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system
type: Item
instance of: scholarly article
title: Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system
zbMATH Open document ID: 1397.49012
DOI: 10.4171/IFB/400
author: Chao-Nien Chen
author: Y. S. Choi
author: Xiaofeng Ren
published in: Interfaces and Free Boundaries
publication date: 2018-08-10T00:00:00Z
review text: Summary: The FitzHugh-Nagumo system gives rise to a geometric variational problem, when its parameters take values in a particular range. A stationary set of the variational problem satisfies an Euler-Lagrange equation that involves the curvature of the boundary of the set and a nonlocal term that inhibits unlimited growth and spreading. The nonlocal term is the solution of a nonhomogeneous modified Helmholtz equation with the characteristic function of the stationary set as the inhomogeneous term. Two types of stationary sets are studied: disc shaped stationary sets in the plane, termed bubbles, and unions of perturbed small discs, termed droplet assemblies, in bounded domains. A complete description of the existence and the stability of bubbles is established. Depending on the parameters, there may be zero, one, two, or even three bubbles. Droplet assemblies are constructed by a reduction argument. Each droplet in a stationary assembly is close in size and shape to the corresponding bubble in the plane. The locations of the droplets in the assembly are determined by the Green's function of the modified Helmholtz equation.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 6915773
zbMATH Keywords: FitzHugh-Nagumo system
zbMATH Keywords: standing pulse
zbMATH Keywords: singular limit
zbMATH Keywords: nonlocal geometric variational problem
zbMATH Keywords: bubbles
zbMATH Keywords: droplet assemblies
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/ifb/400
OpenAlex ID: W2866138816
Wikidata QID: Q129555705
description: scientific article; zbMATH DE number 6915773