On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE
type: Item
instance of: scholarly article
title: On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE
zbMATH Open document ID: 0634.35037
DOI: 10.1016/0022-0396(87)90104-5
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Assuming f satisfies conditions ensuring the infinite dimensional dynamical system defined by the scalar parabolic equation *) \(u_ t=(au_ x)_ x+f(x,u)\), \(x\in (0,1)\), admits a global attractor \({\mathcal A}\), we consider the problem of describing the dynamics of *) restricted to \({\mathcal A}\) by a system of ordinary differential equations. We show that, provided the conductivity function a(x) is near to a limit singular function, such a system of differential equations can be explicitly constructed. Our proof is based on the reduction to a finite dimensional invariant manifold which contains \({\mathcal A}\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 4032210
zbMATH Keywords: structural stability
zbMATH Keywords: infinite dimensional dynamical system
zbMATH Keywords: scalar parabolic equation
zbMATH Keywords: global attractor
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: limit singular function
zbMATH Keywords: invariant manifold
author: Giorgio Fusco
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90104-5
OpenAlex ID: W2014667228
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description: scientific article; zbMATH DE number 4032210