Pattern formation in a nonlinear model for animal coats
type: Item
instance of: scholarly article
title: Pattern formation in a nonlinear model for animal coats
zbMATH Open document ID: 1028.35061
DOI: 10.1016/S0022-0396(02)00156-0
author: Evelyn Sander
author: Thomas Wanner
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The reaction-diffusion system of the form \[ u_t=\varepsilon \Delta u + f(u,v), \quad v_t=d\varepsilon \Delta v + g(u,v) \] is considered; the nonlinearities \(f, g\) satisfy the conditions for the occurence of a Turing instability, e.g. it may be a model of the formation of the animal coat patterns due to Murray: \[ f(u,v)=a-u-h(u,v), \quad g(u,v)=\alpha (b-v) -h(u,v), \] with \(h(u,v)=(1+u+Ku^2)^{-1}\rho uv\). The early stages of the pattern formation process for large domain sizes are described. It is shown that the same mechanism which is responsible for spinodal decomposition in the Cahn-Hilliard equation is responsible for the pattern formation. The authors use techniques of dynamical systems, viewing solutions in terms of nonlinear semiflows.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92B05
zbMATH DE Number: 1915670
zbMATH Keywords: spinodal decomposition
zbMATH Keywords: Turing instability
reviewed by: E. A. Kalita
MaRDI profile type: Publication
cites work: Q4327845
cites work: Maximum norms of chaotic quantum eigenstates and random waves
cites work: Free Energy of a Nonuniform System. I. Interfacial Free Energy
cites work: Q4001895
cites work: Q3764747
cites work: Spinodal decomposition for the cahn-hilliard equation
cites work: Q4039920
cites work: Geometric theory of semilinear parabolic equations
cites work: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces
cites work: Q5628582
cites work: Q4524861
cites work: Spinodal decomposition for multicomponent Cahn-Hilliard systems
cites work: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. I: Probability and wavelength estimate
cites work: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics
cites work: Q4274312
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4000342
cites work: Monte Carlo simulations for spinodal decomposition
cites work: Unexpectedly Linear Behavior for the Cahn--Hilliard Equation
cites work: Infinite-dimensional dynamical systems in mechanics and physics.
cites work: The chemical basis of morphogenesis
cites work: Q3706042
full work available at URL: s0022-0396(02)00156-0
OpenAlex ID: W2165190032
description: scientific article; zbMATH DE number 1915670