Do global attractors depend on boundary conditions?
type: Item
instance of: scholarly article
title: Do global attractors depend on boundary conditions?
zbMATH Open document ID: 0855.58053
author: B. Fiedler
published in: Documenta Mathematica
publication date: 1996-08-05T00:00:00Z
full work available at URL: https://eudml.org/doc/223171
full work available at URL: 11.html
review text: The paper deals with the scalar partial differential equations \[ u_t= u_{xx}+ f(x, u, u_x) \tag{1} \] on the unit interval \(0\leq x\leq 1\) with boundary conditions \[ (1- a_i) u+ a_i u_x= 0, \quad i=0, 1, \qquad \text{at } \quad x=0,\;x=1 \tag{2} \] and where \(0\leq a_i\leq 1\), \(i=0, 1\). For the nonlinearities \(f\in C^2\), such dissipation conditions are imposed that guarantee the local semiflow of \(x\)-profiles of solutions \(u(t, \cdot)\in X^a\), \(t\geq 0\) to be globally defined and dissipative. The state space \(X^a\) is the Sobolev space \(H^2\) intersected with boundary conditions (2), \(a= (a_0, a_1)\). The system \((1)+ (2)\) possesses a global attractor \(A^a_f \subset X^a\) which is the principal object of the author's investigation. Global attractors are called orbit equivalent, if there exists a homeomorphism between them which maps orbits to orbits. For given boundary conditions \(a\), the author defines the attractor class \(A(a)\) as the set of all orbit equivalence classes of global attractors \(A^a_f\). Here \(f\in C^2\) are assumed to be dissipative with only hyperbolic equilibria. The author proves: The global attractor class \(A (a)\) does not depend on the boundary conditions (2) given by \(a= (a_0, a_1)\). The proof is based on the Morse-Smale property and a geometric criterion for hyperbolicity of the set of equilibrium solutions \(u_t= 0\) of \((1)+(2)\). This interesting paper is well written and organized.
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 913529
zbMATH Keywords: global attractor
zbMATH Keywords: orbit equivalence
zbMATH Keywords: boundary conditions
zbMATH Keywords: dissipation conditions
reviewed by: A. Klíč
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 913529