• Previous Article
    The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions
  • CPAA Home
  • This Issue
  • Next Article
    Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces
May  2014, 13(3): 1141-1165. doi: 10.3934/cpaa.2014.13.1141

Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

1. 

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil

2. 

BCAM Basque Center for Applied Mathematics, Mazarredo 14, E-48009 Bilbao, Basque Country

Received  April 2013 Revised  September 2013 Published  December 2013

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.
Citation: Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier, Amsterdam, 2003.

[2]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent , Comm. Partial Differential Equations, 17 (1992), 841-866. doi: 10.1080/03605309208820866.

[3]

T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes , Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci., 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results , Commun. Pure and Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[8]

R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations , J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053.

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[10]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[11]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$ , C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[12]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems , Proc. R. Soc. Edinburgh Sect. A, 135A (2005), 703-730. doi: 10.1017/S030821050000408X.

[13]

M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems , J. Math. Soc. Japan, 63 (2011), 647-673. doi: 10.2969/jmsj/06320647.

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988.

[15]

A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces , Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364.

[16]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors , Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.

[17]

J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations , Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322.

[18]

J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations , Stoch. Dyn., 4 (2004), 385-404. doi: 10.1142/S0219493704001127.

[19]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems , Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.

[20]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces , Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300.

[21]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Appl. Math. Sci. 68, Springer Verlag, New York, 1997.

[24]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin-Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier, Amsterdam, 2003.

[2]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent , Comm. Partial Differential Equations, 17 (1992), 841-866. doi: 10.1080/03605309208820866.

[3]

T. Caraballo, A. N. Carvalho, J. A. Langa and L. F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes , Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci., 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[5]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results , Commun. Pure and Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047.

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[8]

R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations , J. Math. Anal. Appl., 381 (2011), 748-765. doi: 10.1016/j.jmaa.2011.03.053.

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[10]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[11]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\R ^3$ , C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[12]

M. A. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems , Proc. R. Soc. Edinburgh Sect. A, 135A (2005), 703-730. doi: 10.1017/S030821050000408X.

[13]

M. A. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors non-autonomous dissipative systems , J. Math. Soc. Japan, 63 (2011), 647-673. doi: 10.2969/jmsj/06320647.

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988.

[15]

A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces , Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364.

[16]

J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors , Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329.

[17]

J. A. Langa, J. C. Robinson and A. Suárez, Stability, instability and bifurcation phenomena in non-autonomous differential equations , Nonlinearity, 15 (2002), 887-903. doi: 10.1088/0951-7715/15/3/322.

[18]

J. A. Langa and B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations , Stoch. Dyn., 4 (2004), 385-404. doi: 10.1142/S0219493704001127.

[19]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems , Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.

[20]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces , Trans. Amer. Math. Soc., 278 (1983), 21-55. doi: 10.2307/1999300.

[21]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Appl. Math. Sci. 68, Springer Verlag, New York, 1997.

[24]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin-Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

[1]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[2]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[3]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[4]

Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111

[5]

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

[6]

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

[7]

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

[8]

Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021276

[9]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[10]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[11]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

[12]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[13]

Peter E. Kloeden, José Real, Chunyou Sun. Robust exponential attractors for non-autonomous equations with memory. Communications on Pure and Applied Analysis, 2011, 10 (3) : 885-915. doi: 10.3934/cpaa.2011.10.885

[14]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172

[15]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[16]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6267-6284. doi: 10.3934/dcdsb.2021018

[17]

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

[18]

Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087

[19]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085

[20]

Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 73-108. doi: 10.3934/dcds.2021108

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (9)

[Back to Top]