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Title: Finite-dimensional limiting dynamics for dissipative parabolic equations

Abstract

For a broad class of semilinear parabolic equations with compact attractor A in a Banach space E the problem of a description of the limiting phase dynamics (the dynamics on A) of a corresponding system of ordinary differential equations in R{sup N} is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold M subset of E. Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on A; b) the phase semiflow extends on A to a Lipschitz flow; c) the attractor A has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.

Authors:
 [1]
  1. All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences, Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
21202922
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 191; Journal Issue: 3; Other Information: DOI: 10.1070/SM2000v191n03ABEH000466; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; BANACH SPACE; DIFFERENTIAL EQUATIONS; TOPOLOGY; VECTOR FIELDS

Citation Formats

Romanov, A V. Finite-dimensional limiting dynamics for dissipative parabolic equations. United States: N. p., 2000. Web. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Romanov, A V. Finite-dimensional limiting dynamics for dissipative parabolic equations. United States. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Romanov, A V. Sun . "Finite-dimensional limiting dynamics for dissipative parabolic equations". United States. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21202922,
title = {Finite-dimensional limiting dynamics for dissipative parabolic equations},
author = {Romanov, A V},
abstractNote = {For a broad class of semilinear parabolic equations with compact attractor A in a Banach space E the problem of a description of the limiting phase dynamics (the dynamics on A) of a corresponding system of ordinary differential equations in R{sup N} is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold M subset of E. Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on A; b) the phase semiflow extends on A to a Lipschitz flow; c) the attractor A has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.},
doi = {10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 3,
volume = 191,
place = {United States},
year = {2000},
month = {4}
}