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August  2019, 24(8): 3557-3567. doi: 10.3934/dcdsb.2018278

On asymptotically autonomous dynamics for multivalued evolution problems

Jacson Simsen , and  Mariza Stefanello Simsen

Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 - Itajubá - Minas Gerais, Brazil

* Corresponding author: Jacson Simsen, jacson@unifei.edu.br

Dedicated to Peter E. Kloeden on occasion of his 70th birthday

Received  February 2018 Revised  June 2018 Published  October 2018

Fund Project: This work has been partially supported by FAPEMIG (Brazil) - processes PPM 00329-16 and CEX-APQ-00814-16.

In this work we improve the result presented by Kloeden-Simsen-Stefanello Simsen in [8] by reducing uniform conditions. We prove theoretical results in order to establish convergence in the Hausdorff semi-distance of the component subsets of the pullback attractor of a non-autonomous multivalued problem to the global attractor of the corresponding autonomous multivalued problem.

Keywords: Multivalued problem, pullback attractors, asymptotically autonomous dynamics, variable exponents.
Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B41, 37L30.
Citation: Jacson Simsen, Mariza Stefanello Simsen. On asymptotically autonomous dynamics for multivalued evolution problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3557-3567. doi: 10.3934/dcdsb.2018278
References:
[1]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

T. Caraballo, P. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[3]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[4]

T. Caraballo, P. Marin-Rubio and J. C. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour, Set-Valued Analysis, 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[5]

J. I. Díaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.  doi: 10.1006/jmaa.1994.1443.  Google Scholar

[6]

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

[7]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasilinear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[8]

P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[9]

Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[10]

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

[11]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Analysis, 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar

[12]

N. S. Papageorgiou and F. Papalini, On the structure of the solution set of evolution inclusions with time-dependent subdifferentials, Rend. Sem. Mat. Univ. Padova, 97 (1997), 163-186.   Google Scholar

[13]

J. Simsen and E. Capelato, Some properties for exact generalized processes. Continuous and distributed systems. Ⅱ, 209-219, Stud. Syst. Decis. Control, 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[14]

J. Simsen and C. B. Gentile, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124.  doi: 10.1007/s11228-006-0037-1.  Google Scholar

show all references

References:
[1]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

T. Caraballo, P. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[3]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[4]

T. Caraballo, P. Marin-Rubio and J. C. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour, Set-Valued Analysis, 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[5]

J. I. Díaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.  doi: 10.1006/jmaa.1994.1443.  Google Scholar

[6]

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

[7]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasilinear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[8]

P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[9]

Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[10]

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

[11]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Analysis, 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar

[12]

N. S. Papageorgiou and F. Papalini, On the structure of the solution set of evolution inclusions with time-dependent subdifferentials, Rend. Sem. Mat. Univ. Padova, 97 (1997), 163-186.   Google Scholar

[13]

J. Simsen and E. Capelato, Some properties for exact generalized processes. Continuous and distributed systems. Ⅱ, 209-219, Stud. Syst. Decis. Control, 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[14]

J. Simsen and C. B. Gentile, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124.  doi: 10.1007/s11228-006-0037-1.  Google Scholar

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