On the existence of an asymptotically large number of stable dissipative patterns in parabolic systems with small diffusion
type: Item
instance of: scholarly article
title: On the existence of an asymptotically large number of stable dissipative patterns in parabolic systems with small diffusion
zbMATH Open document ID: 0962.35081
DOI: 10.1007/BF02366137
author: A. Yu. Kolesov
author: N. Kh. Rozov
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-11-13T00:00:00Z
review text: The authors study the characteristics of the dynamics of dissipative patterns in the system with small diffusion \[ \frac{\partial u}{\partial t}=\nu D\frac{\partial^2 u}{\partial c^2} +Au+F(u),\quad \frac{\partial u}{\partial x}\Biggl|_{x=0}=\frac{\partial u}{\partial x}|_{x=\pi}=0. \] Here \(u\in \mathbb{R}^2, D\) and \(A\) are Hurwit z matrices, \(\nu>0\) is a parameter responsible for the proportional chance of the diffusion coefficients, \(F\in C^\infty(\mathbb{R}^2,\mathbb{R}^2).\) It is assumed that \(u=0 \) is the unique space-homogeneous equilibrium state, globally stable in the framework of the point model \(\frac{\partial u}{\partial t}=Au+F(u).\) The space of initial conditions (phase space) of the system is \(C([0, \pi];\mathbb{R}^2).\) It is shown that there exist many dissipative patterns, some of them are stable.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1526521
zbMATH Keywords: nonlinear system of reaction-diffusion type
zbMATH Keywords: boundary value problem
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
Wikidata QID: Q59448694
cites work: The chemical basis of morphogenesis
cites work: Q4138569
cites work: BIFURCATION OF INVARIANT TORI OF PARABOLIC SYSTEMS WITH SMALL DIFFUSION
full work available at URL: https://doi.org/10.1007/bf02366137
OpenAlex ID: W1994281704
description: scientific article; zbMATH DE number 1526521