American Institute of Mathematical Sciences

Advanced
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

[1]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[2]

Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009
[3]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[4]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[5]

Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026
[6]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[7]

Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449
[8]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[9]

Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338
[10]

Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270
[11]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
[12]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[13]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
[14]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[15]

Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029
[16]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[17]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[18]

Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351
[19]

Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure & Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687
[20]

James C. Robinson. Inertial manifolds with and without delay. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 813-824. doi: 10.3934/dcds.1999.5.813

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences