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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 5, Pages 129–152 (Mi izv359)  

This article is cited in 7 scientific papers (total in 7 papers)

Finite-dimensional dynamics on attractors of non-linear parabolic equations

A. V. Romanov

All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences

Abstract: We show that one-dimensional semilinear second-order parabolic equations have finite-dimensional dynamics on attractors. In particular, this is true for reaction-diffusion equations with convection on $(0,1)$.
We obtain new topological criteria for a class of dissipative equations of parabolic type in Banach spaces to have finite-dimensional dynamics on invariant compact sets. The dynamics of these equations on an attractor $\mathcal A$ is finite-dimensional (can be described by an ordinary differential equation) if $\mathcal A$ can be embedded in a finite-dimensional $C^1$-submanifold of the phase space.

DOI: https://doi.org/10.4213/im359

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English version:
Izvestiya: Mathematics, 2001, 65:5, 977–1001

Bibliographic databases:

MSC: 37L30, 35B41, 35K57, 35K55, 35B40, 34G20, 35G10, 35K25
Received: 20.07.2000

Citation: A. V. Romanov, “Finite-dimensional dynamics on attractors of non-linear parabolic equations”, Izv. RAN. Ser. Mat., 65:5 (2001), 129–152; Izv. Math., 65:5 (2001), 977–1001

Citation in format AMSBIB
\Bibitem{Rom01}
\by A.~V.~Romanov
\paper Finite-dimensional dynamics on attractors of non-linear parabolic equations
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 5
\pages 129--152
\mathnet{http://mi.mathnet.ru/izv359}
\crossref{https://doi.org/10.4213/im359}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1874356}
\zmath{https://zbmath.org/?q=an:1026.37064}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 5
\pages 977--1001
\crossref{https://doi.org/10.1070/IM2001v065n05ABEH000359}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33747018155}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Romanov, “Effective finite parametrization in phase spaces of parabolic equations”, Izv. Math., 70:5 (2006), 1015–1029  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. A. Eden, S. V. Zelik, V. K. Kalantarov, “Counterexamples to regularity of Mañé projections in the theory of attractors”, Russian Math. Surveys, 68:2 (2013), 199–226  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327  crossref  mathscinet  zmath  isi  scopus
    4. de Moura E.P., Robinson J.C., “Log-Lipschitz Continuity of the Vector Field on the Attractor of Certain Parabolic Equations”, Dyn. Partial Differ. Equ., 11:3 (2014), 211–228  crossref  mathscinet  zmath  isi  scopus
    5. Anna Kostianko, Sergey Zelik, “Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions”, CPAA, 14:5 (2015), 2069  crossref  mathscinet  zmath  scopus
    6. Kostianko A. Zelik S., “Inertial Manifolds For 1D Reaction-Diffusion-Advection Systems. Part i: Dirichlet and Neumann Boundary Conditions”, Commun. Pure Appl. Anal, 16:6 (2017), 2357–2376  crossref  mathscinet  zmath  isi  scopus
    7. Kostianko A. Zelik S., “Nertial Manifolds For 1D Reaction-Diffusion-Advection Systems. Part II: Periodic Boundary Conditions”, Commun. Pure Appl. Anal, 17:1 (2018), 285–317  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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