August  2016, 9(4): 979-994. doi: 10.3934/dcdss.2016037

Structure of the pullback attractor for a non-autonomous scalar differential inclusion

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain, Spain

2. 

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. Universidad, s/n, 03202-Elche, Spain

Received  September 2015 Revised  January 2016 Published  August 2016

The structure of attractors for differential equations is one of the main topics in the qualitative theory of dynamical systems. However, the theory is still in its infancy in the case of multivalued dynamical systems. In this paper we study in detail the structure and internal dynamics of a scalar differential equation, both in the autonomous and non-autonomous cases. To this aim, we will also show a general result on the characterization of a pullback attractor for a multivalued process by the union of all the complete bounded trajectories of the system.
Citation: T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
References:
[1]

J. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations,, J. Differential Equations, 27 (1978), 224.  doi: 10.1016/0022-0396(78)90032-3.  Google Scholar

[2]

E. Capelato and J. Simsen, Some properties for exact generalized processes,, in Continuous and Distributed Systems II (V.A Zadovnichiy and M.Z. Zgurovsky eds.), 30 (2015), 209.  doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[3]

T. Caraballo, J. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems,, Dyn. System: An Int. J., 20 (2005), 301.  doi: 10.1080/14689360500151847.  Google Scholar

[4]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Anal., 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[5]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems,, Springer, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[6]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour,, Set-Valued Anal., 11 (2003), 297.  doi: 10.1023/A:1024422619616.  Google Scholar

[7]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors,, SIAM J. Math. Anal., 47 (2015), 1530.  doi: 10.1137/140978995.  Google Scholar

[8]

M. O. Gluzman, N. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications,, Appl. Math. Lett., 39 (2015), 19.  doi: 10.1016/j.aml.2014.08.006.  Google Scholar

[9]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system,, J. Math. Anal. Appl., 373 (2011), 535.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[10]

O. V. Kapustyan, O. P. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term,, Discrete and Continuous Dynamical Systems, 34 (2014), 4155.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[11]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feeback control problem,, Set-Valued Var. Anal., 21 (2013), 271.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

[12]

J. A. Langa, J. C. Robinson and A. Suarez, Stability, instability, and bifurcation phenomena in nonautonomous differential equations,, Nonlinearity, 15 (2002), 887.  doi: 10.1088/0951-7715/15/3/322.  Google Scholar

[13]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar

[14]

A. Rodrígez-Bernal and A. Vidal-López, Existence, uniqueness and attractivity properties of positive complete trajectories for nonautonomous reaction-diffusion problems,, Discrete and Continuous Dynamical Systems, 18 (2007), 537.  doi: 10.3934/dcds.2007.18.537.  Google Scholar

[15]

M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in nonsmooth systems with applications for earth data processing: uniform trajectory attractors for nonautonomous evolution inclusions solutions with pointwise pseudomonotone mappings,, in Advances in global optimization, 95 (2015), 283.  doi: 10.1007/978-3-319-08377-3_28.  Google Scholar

[16]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III,, Springer, (2012).  doi: 10.1007/978-3-642-28512-7.  Google Scholar

show all references

References:
[1]

J. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations,, J. Differential Equations, 27 (1978), 224.  doi: 10.1016/0022-0396(78)90032-3.  Google Scholar

[2]

E. Capelato and J. Simsen, Some properties for exact generalized processes,, in Continuous and Distributed Systems II (V.A Zadovnichiy and M.Z. Zgurovsky eds.), 30 (2015), 209.  doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[3]

T. Caraballo, J. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems,, Dyn. System: An Int. J., 20 (2005), 301.  doi: 10.1080/14689360500151847.  Google Scholar

[4]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Anal., 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[5]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems,, Springer, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[6]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour,, Set-Valued Anal., 11 (2003), 297.  doi: 10.1023/A:1024422619616.  Google Scholar

[7]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors,, SIAM J. Math. Anal., 47 (2015), 1530.  doi: 10.1137/140978995.  Google Scholar

[8]

M. O. Gluzman, N. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications,, Appl. Math. Lett., 39 (2015), 19.  doi: 10.1016/j.aml.2014.08.006.  Google Scholar

[9]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system,, J. Math. Anal. Appl., 373 (2011), 535.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[10]

O. V. Kapustyan, O. P. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term,, Discrete and Continuous Dynamical Systems, 34 (2014), 4155.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[11]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feeback control problem,, Set-Valued Var. Anal., 21 (2013), 271.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

[12]

J. A. Langa, J. C. Robinson and A. Suarez, Stability, instability, and bifurcation phenomena in nonautonomous differential equations,, Nonlinearity, 15 (2002), 887.  doi: 10.1088/0951-7715/15/3/322.  Google Scholar

[13]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar

[14]

A. Rodrígez-Bernal and A. Vidal-López, Existence, uniqueness and attractivity properties of positive complete trajectories for nonautonomous reaction-diffusion problems,, Discrete and Continuous Dynamical Systems, 18 (2007), 537.  doi: 10.3934/dcds.2007.18.537.  Google Scholar

[15]

M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in nonsmooth systems with applications for earth data processing: uniform trajectory attractors for nonautonomous evolution inclusions solutions with pointwise pseudomonotone mappings,, in Advances in global optimization, 95 (2015), 283.  doi: 10.1007/978-3-319-08377-3_28.  Google Scholar

[16]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III,, Springer, (2012).  doi: 10.1007/978-3-642-28512-7.  Google Scholar

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[3]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[4]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[5]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[6]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[7]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[8]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

[9]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[10]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[11]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[12]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[13]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[14]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[15]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[16]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[17]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[18]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[19]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[20]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (71)
  • HTML views (1)
  • Cited by (2)

Other articles
by authors

[Back to Top]