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

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

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

T. Caraballo ^1, , J. A. Langa ^1, and J. Valero ^2,

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

Abstract
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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.

Keywords: structure of the attracor, Non-autonomous dynamical systems, differential inclusion, pullback attractor, multivalued dynamical systems..

Mathematics Subject Classification: 35B40, 35B41, 34C37, 34D4.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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