# American Institute of Mathematical Sciences

2013, 33(2): 483-503. doi: 10.3934/dcds.2013.33.483

## Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay

 1 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 2 The Academy of Journalism and Communication, 36 Xuan Thuy, Cau Giay, Hanoi, Vietnam 3 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam

Received  July 2011 Revised  April 2012 Published  September 2012

Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear parabolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive point spectra, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.
Citation: Cung The Anh, Le Van Hieu, Nguyen Thieu Huy. Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 483-503. doi: 10.3934/dcds.2013.33.483
##### References:
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##### References:
 [1] A. Bensoussan and F. Landoli, Stochastic inertial manifolds,, Stochastics Rep., 53 (1995), 13. [2] L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations,, Nonlinear Anal., 34 (1998), 907. doi: 10.1016/S0362-546X(97)00569-5. [3] A. P. Calderon, Spaces between $L^1$ and $L^\infty$ and the theorem of Marcinkiewicz,, Studia Math, 26 (1996), 273. [4] T. Caraballo and J. A. Langa, Tracking properties of trajectories on random attracting sets,, Stochastic Anal. Appl., 17 (1999), 339. doi: 10.1080/07362999908809605. [5] I. D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise,, J. Dyn. Differ. Equations, 7 (1995), 549. [6] I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (2002). [7] I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dyn. Differ. Equations, 13 (2001), 355. [8] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations,, J. Differ. Equations, 73 (1988), 309. doi: 10.1016/0022-0396(88)90110-6. [9] A. Y. Goritskij and M. I. Vishik, Local integral manifolds for a nonautonomous parabolic equation,, J. Math. Sci., 85 (1997), 2428. doi: 10.1007/BF02355848. [10] N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. [11] N. T. Huy, Inertial manifolds for semilinear parabolic equations in admissible spaces,, J. Math. Anal. Appl., 386 (2012), 894. doi: 10.1016/j.jmaa.2011.08.051. [12] N. Koksch and S. Siegmund, Pullback attracting inertial manifols for nonautonomous dynamical systems,, J. Dyn. Differ. Equations, 14 (2002), 889. [13] Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds,, Discrete and Continuous Dynamical System, 5 (1999), 233. doi: 10.3934/dcds.1999.5.233. [14] J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces II, Function Spaces,", Springer-Verlag, (1979). [15] J. J. Massera and J. J. Schäffer, "Linear Differential Equations and Function Spaces,", Academic Press, (1966). [16] M. Taboado and Y. You, Invariant manifolds for retarded semilinear wave equations,, J. Differ. Equations, 114 (1994), 337. doi: 10.1006/jdeq.1994.1153. [17] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer, (2002). [18] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, (1997).
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