A minimization problem associated with elliptic systems of FitzHugh--Nagumo type
type: Item
instance of: scholarly article
title: A minimization problem associated with elliptic systems of FitzHugh--Nagumo type
zbMATH Open document ID: 1110.35019
author: Shusen Yan
author: E. N. Dancer
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2004-11-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78617
review text: The authors deal with the following problem: \[ \begin{cases} -\varepsilon^2\Delta u=f(u)-v\quad & \text{in }\Omega\\ -\Delta v+ \gamma v=\delta u\quad &\text{in }\Omega\\ u=v=0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\varepsilon\) is a parameter, \(\gamma\) and \(\delta\) are nonnegative constants, and \(f\) is a \(C^1\)-function satisfying some natural conditions. The aim of the authors is to prove that for suitably large \(\delta>0\), (1) has solutions, which either oscillate around a constant in a compact subset of \(\Omega\), or have a sharp interior layer. These solutions are local minimum of the corresponding functional.
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 2117903
zbMATH Keywords: Elliptic systems
zbMATH Keywords: Multiple layer
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
OpenAlex ID: W2058421769
cites work: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property
cites work: Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi
cites work: Q4538002
cites work: Q5665783
cites work: A stability analysis for a semi-linear parabolic partial differential equation
cites work: On a nonlinear eigenvalue problem occurring in population genetics
cites work: Q3811030
cites work: On the Number of Positive Solutions of Weakly Non-Linear Elliptic Equations when a Parameter is Large
cites work: Q4191164
cites work: A Maximum Principle for an Elliptic System and Applications to Semilinear Problems
cites work: On a conjecture of De Giorgi and some related problems
cites work: On De Giorgi's conjecture in dimensions 4 and 5
cites work: Q4040640
cites work: Standing Wave Solutions for a System Derived from the Fitzhugh–Nagumo Equations for Nerve Conduction
cites work: Stationary Wave Solutions of a System of Reaction-Diffusion Equations Derived from the FitzHugh–Nagumo Equations
cites work: On steady state solutions of a system of reaction-diffusion equations from biology
cites work: The gradient theory of phase transitions and the minimal interface criterion
cites work: A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type
cites work: A positive solution on \(\mathbb{R}^N\) to a system of elliptic equations of FitzHugh-Nagumo type
cites work: Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system.
cites work: Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type
cites work: The effect of a singular perturbation on nonconvex variational problems
DOI: 10.1016/J.ANIHPC.2003.02.001
description: scientific article; zbMATH DE number 2117903