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Mat. Sb., 2005, Volume 196, Number 4, Pages 23–50 (Mi msb1282)  

This article is cited in 11 scientific papers (total in 11 papers)

Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds

A. Yu. Goritskiia, V. V. Chepyzhovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary differential equation $du/dt+Au=F(u,t)$ in a Hilbert space $H$. It is assumed that the non-linear function $F(u,t)$ is essentially subordinated to the linear operator $A$; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds $M_+$ and $M_-$ attracting at an exponential rate an arbitrary solution of this equation as $t\to+\infty$ and $t\to-\infty$, respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type.

DOI: https://doi.org/10.4213/sm1282

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English version:
Sbornik: Mathematics, 2005, 196:4, 485–511

Bibliographic databases:

UDC: 517.956
MSC: Primary 34G20, 34C45, 35B42, 35G10; Secondary 35K57
Received: 25.04.2004

Citation: A. Yu. Goritskii, V. V. Chepyzhov, “Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds”, Mat. Sb., 196:4 (2005), 23–50; Sb. Math., 196:4 (2005), 485–511

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chepyzhov V.V., Goritsky A.Yu., Vishik M.I., “Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation”, Russ. J. Math. Phys., 12:1 (2005), 17–39  mathscinet  zmath  isi  elib
    2. 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
    3. Koksch N., Siegmund S., “Feedback control via inertial manifolds for nonautonomous evolution equations”, Commun. Pure Appl. Anal., 10:3 (2009), 917–936  crossref  mathscinet  isi
    4. Chalkina N.A., “Inertsialnoe mnogoobrazie dlya giperbolicheskogo uravneniya s dissipatsiei”, Vestnik moskovskogo universiteta. seriya 1: matematika. mekhanika, 2011, no. 6, 3–7  mathnet  mathscinet  zmath  elib
    5. Chalkina N.A., “Sufficient Condition for the Existence of an Inertial Manifold for a Hyperbolic Equation with Weak and Strong Dissipation”, Russ. J. Math. Phys., 19:1 (2012), 11–20  crossref  mathscinet  zmath  isi  elib
    6. 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
    7. A. Yu. Goritskii, N. A. Chalkina, “Inertial manifolds for weakly and strongly dissipative hyperbolic equations”, J. Math. Sci. (N. Y.), 197:3 (2014), 291–302  mathnet  crossref  elib
    8. 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  elib
    9. Pimentel E.A., Pimentel J.F.S., “Estimates for a class of slowly non-dissipative reaction-diffusion equations”, Rocky Mt. J. Math., 46:3 (2016), 1011–1028  crossref  mathscinet  zmath  isi  scopus
    10. Chepyzhov V.V., Kostianko A., Zelik S., “Inertial Manifolds For the Hyperbolic Relaxation of Semilinear Parabolic Equations”, Discrete Contin. Dyn. Syst.-Ser. B, 24:3, SI (2019), 1115–1142  crossref  mathscinet  zmath  isi  scopus
    11. Li X. Sun Ch., “Inertial Manifolds For the 3D Modified-Leray-Alpha Model”, J. Differ. Equ., 268:4 (2020), 1532–1569  crossref  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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