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Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 4, Pages 36–54 (Mi izv855)  

This article is cited in 15 scientific papers (total in 15 papers)

Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations

A. V. Romanov

Abstract: Sufficient conditions are obtained for the existence of a $k$-dimensional invariant manifold that attracts as $t\to\infty$ all solutions $u(t)$ of the evolution equation $\dot u=-Au+F(u)$ in a Hilbert space, where $A$ is a linear selfadjoint operator, semibounded from below, with compact resolvent, and $F$ is a uniformly Lipschitz (in suitable norms) nonlinearity; these conditions sharpen previously known conditions and cannot be improved.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 43:1, 31–47

Bibliographic databases:

UDC: 517.95
MSC: Primary 34G20, 35K22, 47H15, 58F39; Secondary 35K57, 58F12
Received: 21.06.1991

Citation: A. V. Romanov, “Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations”, Izv. RAN. Ser. Mat., 57:4 (1993), 36–54; Russian Acad. Sci. Izv. Math., 43:1 (1994), 31–47

Citation in format AMSBIB
\by A.~V.~Romanov
\paper Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations
\jour Izv. RAN. Ser. Mat.
\yr 1993
\vol 57
\issue 4
\pages 36--54
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 43
\issue 1
\pages 31--47

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    This publication is cited in the following articles:
    1. T. V. Girya, I. D. Chueshov, “Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems”, Sb. Math., 186:1 (1995), 29–45  mathnet  crossref  mathscinet  zmath  isi
    2. A. V. Romanov, “On the limit dynamics of evolution equations”, Russian Math. Surveys, 51:2 (1996), 345–346  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. A. V. Romanov, “Three counterexamples in the theory of inertial manifolds”, Math. Notes, 68:3 (2000), 378–385  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. V. Romanov, “Finite-dimensional limiting dynamics for dissipative parabolic equations”, Sb. Math., 191:3 (2000), 415–429  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. A. V. Romanov, “Finite-dimensional dynamics on attractors of non-linear parabolic equations”, Izv. Math., 65:5 (2001), 977–1001  mathnet  crossref  crossref  mathscinet  zmath
    6. A. Yu. Goritskii, V. V. Chepyzhov, “Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds”, Sb. Math., 196:4 (2005), 485–511  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Lingli Xie, Kok-lay Teo, Yi Zhao, “Chaos synchronization for continuous chaotic systems by inertial manifold approach”, Chaos, Solitons & Fractals, 32:1 (2007), 234  crossref  elib
    8. Yu. A. Goritsky, “Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation”, J. Math. Sci. (N. Y.), 143:4 (2007), 3239–3252  mathnet  crossref  mathscinet  elib
    9. Igor Chueshov, Björn Schmalfuß, “Master-slave synchronization and invariant manifolds for coupled stochastic systems”, J Math Phys (N Y ), 51:10 (2010), 102702  crossref  elib
    10. 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
    11. A. V. Romanov, “A Parabolic Equation with Nonlocal Diffusion without a Smooth Inertial Manifold”, Math. Notes, 96:4 (2014), 548–555  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327  crossref  isi
    13. Anna Kostianko, Sergey Zelik, “Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions”, CPAA, 14:5 (2015), 2069  crossref
    14. 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  isi
    15. 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  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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