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

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

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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• http://mi.mathnet.ru/eng/msb1282
• https://doi.org/10.4213/sm1282
• http://mi.mathnet.ru/eng/msb/v196/i4/p23

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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. 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
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
3. Koksch N., Siegmund S., “Feedback control via inertial manifolds for nonautonomous evolution equations”, Commun. Pure Appl. Anal., 10:3 (2009), 917–936
4. Chalkina N.A., “Inertsialnoe mnogoobrazie dlya giperbolicheskogo uravneniya s dissipatsiei”, Vestnik moskovskogo universiteta. seriya 1: matematika. mekhanika, 2011, no. 6, 3–7
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
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
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
8. Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327
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
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
11. Li X. Sun Ch., “Inertial Manifolds For the 3D Modified-Leray-Alpha Model”, J. Differ. Equ., 268:4 (2020), 1532–1569
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