# Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay

Alexander V. Rezounenko^{[1]}

- [1] Kharkov University, Department of Mechanics and Mathematics, 4 Svobody sqr., Kharkov 61077 (Ukraine)

Annales de l’institut Fourier (2004)

- Volume: 54, Issue: 5, page 1547-1564
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topRezounenko, Alexander V.. "Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay." Annales de l’institut Fourier 54.5 (2004): 1547-1564. <http://eudml.org/doc/116151>.

@article{Rezounenko2004,

abstract = {Inertial manifold with delay (IMD) for dissipative systems of second order in time is
constructed. This result is applied to the study of different asymptotic properties of
solutions. Using IMD, we construct approximate inertial manifolds containing all the
stationary solutions and give a new characterization of the K-invariant manifold.},

affiliation = {Kharkov University, Department of Mechanics and Mathematics, 4 Svobody sqr., Kharkov 61077 (Ukraine)},

author = {Rezounenko, Alexander V.},

journal = {Annales de l’institut Fourier},

keywords = {approximate inertial manifolds; inertial manifolds with delay; retarded non-linear partial differential equations; retarded nonlinear partial differential equations},

language = {eng},

number = {5},

pages = {1547-1564},

publisher = {Association des Annales de l'Institut Fourier},

title = {Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay},

url = {http://eudml.org/doc/116151},

volume = {54},

year = {2004},

}

TY - JOUR

AU - Rezounenko, Alexander V.

TI - Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay

JO - Annales de l’institut Fourier

PY - 2004

PB - Association des Annales de l'Institut Fourier

VL - 54

IS - 5

SP - 1547

EP - 1564

AB - Inertial manifold with delay (IMD) for dissipative systems of second order in time is
constructed. This result is applied to the study of different asymptotic properties of
solutions. Using IMD, we construct approximate inertial manifolds containing all the
stationary solutions and give a new characterization of the K-invariant manifold.

LA - eng

KW - approximate inertial manifolds; inertial manifolds with delay; retarded non-linear partial differential equations; retarded nonlinear partial differential equations

UR - http://eudml.org/doc/116151

ER -

## References

top- C. Foias, G. Sell, R. Temam, Variétés Inertielles des équations différentielles dissipatives, C.R. Acad. Sci. Paris, Série I 301 (1985), 139-141 Zbl0591.35062MR801946
- S.-N. Chow, K. Lu, Invariant manifolds for flows in Banach spaces, J. Diff. Eqns 74 (1988), 285-317 Zbl0691.58034MR952900
- C. Foias, O. Manley, R. Temam, Sur l'interaction des petits et grands tourbillons dans les écoulements turbulents, C.R. Acad. Sci. Paris, Série I 305 (1987), 497-500 Zbl0624.76072MR916319
- C. Foias G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34 Zbl0176.54103MR223716
- C. Foias, G. Sell, E. Titi, Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dyn. Diff. Eqns 1 (1989), 199-224 Zbl0692.35053MR1010966
- I.D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems (in Russian), Acta, Kharkov (1999) (2002), http://www.emis.de/monographs/Chueshov/ Zbl1100.37046
- I.D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dyn. Diff. Eqns 7 (1995), 549-566 Zbl0840.60054MR1362670
- A. Debussche, R. Temam, Inertial manifolds and their dimension, (1993), World Scientific Zbl0769.34047MR1386913
- P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, (1989), Springer, Berlin-Heidelberg-New York Zbl0683.58002MR966192
- R. Temam, Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer, Berlin-Heidelberg-New York Zbl0662.35001MR953967
- L. Boutet de Monvel, I.D. Chueshov, A.V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis 34 (1998), 907-925 Zbl0954.34064MR1636608
- X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Res. Notes in Math 155 (1987), 172-183 Zbl0642.35061MR907731
- R. Datko, Y.C. You, Some second-order vibrating systems cannot tolerate small delays in their damping, J. Optim. Theory. Appl 70 (1991), 521-537 Zbl0791.34045MR1124776
- A. Debussche, R. Temam, Some new generalizations of inertial manifolds, Discr. Contin. Dynamical Systems 2 (1996), 543-558 Zbl0948.35018MR1414085
- J. Robinson, Inertial manifolds with and without delay, Discr. Contin. Dynamical Systems 5 (1999), 813-824 Zbl0954.37037MR1722373
- M. Taboada, Y.C. You, Invariant manifolds for retarded semilinear wave equations, J. Diff. Eqns 114 (1994), 337-369 Zbl0815.34067MR1303032
- J.K. Hale, Theory of functional differential equations, (1977), Springer, Berlin - Heidelberg - New York Zbl0352.34001MR508721
- J.K. Hale, Asymptotic behavior of dissipative systems, (1988), Amer. Math. Soc., Providence, Rhode Island Zbl0642.58013MR941371
- C.C. Travis, D.F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS 200 (1974), 395-418 Zbl0299.35085MR382808
- O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay equations. Functional, complex, and nonlinear analysis, 110 (1995), Springer-Verlag, New York Zbl0826.34002MR1345150
- J. Wu, Theory and applications of partial functional-differential equation, 119 (1996), Springer-Verlag, New York Zbl0870.35116
- I.D. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity, J. of Soviet Mathematics 58 (1992), 385-390 Zbl0783.73046
- I.D. Chueshov, A.V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, série I 321 (1995), 607-612 Zbl0845.35129MR1356562
- L. Boutet de Monvel, I.D. Chueshov, and A.V. Rezounenko, Long-time behaviour of strong solutions for a class of retarded nonlinear P.D.E.s, Commun. in Partial Differential Equations 22 (1997), 1453-1474 Zbl0891.35159MR1469578
- A.V. Rezounenko, Inertial manifolds with delay for retarded semilinear parabolic equations, Discr. Contin. Dynamical Systems 6 (2000), 829-840 Zbl1011.37046MR1788255
- A.V. Rezounenko, Inertial manifolds for retarded second order in time evolution equations, Nonlinear Analysis 51 (2002), 1045-1054 Zbl1023.35087MR1926084
- A.V. Rezounenko, Steady approximate inertial manifolds of exponential order for semilinear parabolic equations, Differential and Integral Equations 15 (2002), 1345-1356 Zbl1161.35427MR1920691
- A.V. Rezounenko, Approximate inertial manifolds for retarded semilinear parabolic equations, J. Math. Anal. Appl 282 (2003), 614-628 Zbl1039.35133MR1989676
- M.S. Jolly, I.G. Kevrekidis, E.S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D 44 (1990), 38-60 Zbl0704.58030MR1069671

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.