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 and 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), 224-265. doi: 10.1016/0022-0396(78)90032-3.

[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.), Springer, Cham, 30 (2015), 209-219. doi: 10.1007/978-3-319-19075-4_12.

[3]

T. Caraballo, J. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems, Dyn. System: An Int. J., 20 (2005), 301-321. doi: 10.1080/14689360500151847.

[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), 153-201. doi: 10.1023/A:1022902802385.

[5]

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

[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), 297-322. doi: 10.1023/A:1024422619616.

[7]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995.

[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), 19-21. doi: 10.1016/j.aml.2014.08.006.

[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), 535-547. doi: 10.1016/j.jmaa.2010.07.040.

[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), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[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), 271-282. doi: 10.1007/s11228-013-0233-8.

[12]

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

[13]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.

[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), 537-567. doi: 10.3934/dcds.2007.18.537.

[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, Springer Proc. Math. Stat., Springer, Cham, 95 (2015), 283-294. doi: 10.1007/978-3-319-08377-3_28.

[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, Heidelberg, 2012. doi: 10.1007/978-3-642-28512-7.

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), 224-265. doi: 10.1016/0022-0396(78)90032-3.

[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.), Springer, Cham, 30 (2015), 209-219. doi: 10.1007/978-3-319-19075-4_12.

[3]

T. Caraballo, J. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems, Dyn. System: An Int. J., 20 (2005), 301-321. doi: 10.1080/14689360500151847.

[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), 153-201. doi: 10.1023/A:1022902802385.

[5]

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

[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), 297-322. doi: 10.1023/A:1024422619616.

[7]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995.

[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), 19-21. doi: 10.1016/j.aml.2014.08.006.

[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), 535-547. doi: 10.1016/j.jmaa.2010.07.040.

[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), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[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), 271-282. doi: 10.1007/s11228-013-0233-8.

[12]

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

[13]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.

[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), 537-567. doi: 10.3934/dcds.2007.18.537.

[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, Springer Proc. Math. Stat., Springer, Cham, 95 (2015), 283-294. doi: 10.1007/978-3-319-08377-3_28.

[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, Heidelberg, 2012. doi: 10.1007/978-3-642-28512-7.

[1]

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

[2]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088

[3]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[4]

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

[5]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935

[6]

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

[7]

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

[8]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569

[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]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[19]

Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006

[20]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (177)
  • HTML views (1)
  • Cited by (2)

Other articles
by authors

[Back to Top]