Point-condensation for a reaction-diffusion system
type: Item
instance of: scholarly article
title: Point-condensation for a reaction-diffusion system
zbMATH Open document ID: 0627.35049
DOI: 10.1016/0022-0396(86)90119-1
author: Izumi Takagi
published in: Journal of Differential Equations
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with a system of the form \[ (*)\quad d_ 1u''+f(u,v)=0,\quad d_ 2v''g(u,v)=0\quad on\quad [0,1] \] subject to homogeneous Neumann boundary conditions at 0 and 1. The diffusion coefficients \(d_ 1\) and \(d_ 2\) are positive parameters, here. The problem arises as a normalized steady state system from a reaction- diffusion system proposed by Gierer and Meinhardt. Assuming \(d_ 2\) to be sufficiently large the author shows that the system possesses ``point-condensing'' solutions, if \(d_ 1\) is sufficiently small. Moreover, he obtains the following global result: Depending on whether bifurcation from the constant solutions occurs or not, there exists a continuum either connecting them to the point- condensing solutions or connecting the point-condensing solutions to another family of such solutions.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4019595
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: diffusion coefficients
zbMATH Keywords: steady state system
zbMATH Keywords: point-condensing
zbMATH Keywords: continuum
reviewed by: Georg Hetzer
MaRDI profile type: Publication
full work available at URL: 0022-0396(86)90119-1
OpenAlex ID: W1977010796
cites work: Pattern formation in morphogenesis. Analytical treatment of the Gierer- Meinhardt model on a sphere
cites work: Q3937178
cites work: Bifurcation from simple eigenvalues
cites work: Bifurcation, perturbation of simple eigenvalues, and linearized stability
cites work: Q5110819
cites work: A bifurcation analysis of pattern formation in a diffusion governed morphogenetic field
cites work: Stationary solutions of reaction-diffusion equations
cites work: Synergetics. An introduction. Nonequilibrium phase transitions and self-organization in physics, chemistry, and biology. 3rd rev. and enl. ed
cites work: Bifurcation theory for Fredholm operators
cites work: Activators and Inhibitors in Pattern Formation
cites work: Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter
cites work: Q3879128
cites work: Variational methods in nonlinear problems
cites work: Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems
cites work: Q4742474
cites work: Non-degenerate solutions of boundary-value problems
cites work: Global bifurcation of steady-state solutions
cites work: Stability of bifurcating solutions of the Gierer-Meinhardt system
cites work: A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt
description: scientific article; zbMATH DE number 4019595