American Institute of Mathematical Sciences

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February  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

Cung The Anh 1, , Le Van Hieu 2, and  Nguyen Thieu Huy 3,

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.
Keywords: semilinear parabolic equations, non-autonomous dynamical systems, spectral gap condition, Inertial manifolds, finite delay, admissible function spaces..
Mathematics Subject Classification: 35B40, 35B42, 49K25, 35K55, 34C3.
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:
[1]

A. Bensoussan and F. Landoli, Stochastic inertial manifolds,, Stochastics Rep., 53 (1995), 13.   Google Scholar

[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.  Google Scholar

[3]

A. P. Calderon, Spaces between $L^1$ and $L^\infty$ and the theorem of Marcinkiewicz,, Studia Math, 26 (1996), 273.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (2002).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. Koksch and S. Siegmund, Pullback attracting inertial manifols for nonautonomous dynamical systems,, J. Dyn. Differ. Equations, 14 (2002), 889.   Google Scholar

[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.  Google Scholar

[14]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces II, Function Spaces,", Springer-Verlag, (1979).   Google Scholar

[15]

J. J. Massera and J. J. Schäffer, "Linear Differential Equations and Function Spaces,", Academic Press, (1966).   Google Scholar

[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.  Google Scholar

[17]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer, (2002).   Google Scholar

[18]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, (1997).   Google Scholar

show all references

References:
[1]

A. Bensoussan and F. Landoli, Stochastic inertial manifolds,, Stochastics Rep., 53 (1995), 13.   Google Scholar

[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.  Google Scholar

[3]

A. P. Calderon, Spaces between $L^1$ and $L^\infty$ and the theorem of Marcinkiewicz,, Studia Math, 26 (1996), 273.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (2002).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. Koksch and S. Siegmund, Pullback attracting inertial manifols for nonautonomous dynamical systems,, J. Dyn. Differ. Equations, 14 (2002), 889.   Google Scholar

[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.  Google Scholar

[14]

J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces II, Function Spaces,", Springer-Verlag, (1979).   Google Scholar

[15]

J. J. Massera and J. J. Schäffer, "Linear Differential Equations and Function Spaces,", Academic Press, (1966).   Google Scholar

[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.  Google Scholar

[17]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer, (2002).   Google Scholar

[18]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, (1997).   Google Scholar

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