Structural stability for scalar parabolic equations
type: Item
instance of: scholarly article
title: Structural stability for scalar parabolic equations
zbMATH Open document ID: 0810.35045
author: Kening Lu
published in: Journal of Differential Equations
publication date: 1995-04-10T00:00:00Z
review text: The author considers the parabolic equation \[ u_ t= u_{xx}+ f(u), \quad 0<x<1, \qquad u(t,0)= u(t,1) =0.\tag{1} \] Here \(f\) is smooth and satisfies \(\limsup f(u) u^{-1}=0\) as \(| u| \to\infty\), and \(f(0)=0\). Equation (1) is rewritten as an evolution equation in the form (2) \(u_ t+ Au= \overline{F} (u)\). The underlying space is \(H_ 0^ 1\) with \(\text{dom} (A)=\{u\mid u\in H^ 2\cap H_ 0^ 1\), \(u_{xx}\in H_ 0^ 1\}\) and \(Au=- u_{xx}\) for \(u\in \text{dom}(A)\). Moreover \(\overline{F} (u)= f(u)\) turns out to be a \(C^ 1\) mapping from \(H_ 0^ 1\) into \(H_ 0^ 1\). Equation (2) admits an absorbing ball \(B_ R\) which gives rise to a truncated function \(F(u)= \theta(| u| (2R)^{-1}) \overline{F} (u)\) (\(|\cdot |\) the norm on \(H_ 0^ 1\)) where \(\theta (s)=1\) for \(s\in [0,1]\), \(\theta (s)=0\) for \(s\in [2,\infty)\) and \(|\theta' (s)|\leq 2\) for \(s\geq 0\). Instead of equation (2) the truncated equation (3) \(u_ t+ Au= F(u)\) is now studied. More precisely one considers a perturbation of (1), i.e. a parabolic equation (4) \(u_ t= u_{xx}+ g(u)\), \(0<x<1\) and \(u(t,0)= u(t,1) =0\) with smooth \(g\). Equation (4) is put into an abstract form similar to (2) and then replaced by a corresponding truncated equation assuming the form (5) \(u_ t+ Au= G(u)\). The main result compares (3) with (5). In somewhat imprecise terms it states: If \(F\), \(G\) are close in a certain specific sense then there is a neighbourhood \(U\) of the attractor of (3) and a neighbourhood \(V\) of the attractor of (5) and a homeomorphism \(\varphi\) from \(U\) onto \(V\) which transforms orbits of (3) into orbits of (5) and preserves the direction of time. The theorem thus expresses a property of structural stability of (3). The proof is based on techniques used in inertial manifold theory and used in similar contexts by some of the authors in the reference list.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 58D25
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 696890
zbMATH Keywords: structural stability
zbMATH Keywords: inertial manifold theory
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1150
OpenAlex ID: W2017908968
DOI: 10.1006/JDEQ.1994.1150
description: scientific article; zbMATH DE number 696890