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February  2020, 19(2): 1001-1016. doi: 10.3934/cpaa.2020046

Asymptotic behavior of coupled inclusions with variable exponents

Peter E. Kloeden 1,, , Jacson Simsen 2, and  Petra Wittbold 3,

1. 

Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany
2. 

Instituto de Matemática e Computação, Universidade Federal de Itajubá, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá - MG - Brazil
3. 

Fakultät für Mathematik, Universität of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

* Corresponding author

Received  March 2019 Revised  June 2019 Published  October 2019

Fund Project: This work was initiated when the second author was supported with CNPq scholarship - process 202645/2014-2 (Brazil). The first author was suported by Chinese NSF grant 11571125. The second author was partially supported by the Brazilian research agency FAPEMIG process PPM 00329-16.

This work concerns the study of asymptotic behavior of the solutions of a nonautonomous coupled inclusion system with variable exponents. We prove the existence of a pullback attractor and that the system of inclusions is asymptotically autonomous.

Keywords: Pullback attractor, reaction-diffusion coupled systems, variable exponents, asymptotically autonomous problems.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K57; Secondary: 35K55, 35K92.
Citation: Peter E. Kloeden, Jacson Simsen, Petra Wittbold. Asymptotic behavior of coupled inclusions with variable exponents. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1001-1016. doi: 10.3934/cpaa.2020046
References:
[1]

C. O. Alves, S. Shmarev, J. Simsen and M. S. Simsen, The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: existence and asymptotic behavior, J. Math. Anal. Appl., 443 (2016), 265-294.  doi: 10.1016/j.jmaa.2016.05.024.  Google Scholar

[2]

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

[3]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Berlin, 1990.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[8]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[9]

P. Harjulehto, P. Hästö, U. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Analysis, 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.  Google Scholar

[10]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[11]

P. E. Kloeden and P. Marín-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, J. Setvalued & Variational Analysis, 19 (2011), 43-57.  doi: 10.1007/s11228-009-0123-2.  Google Scholar

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc. Providence, 2011. doi: 10.1090/surv/176.  Google Scholar

[13]

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

[14]

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

[15]

P. E. Kloeden, J. Simsen and M. S. Simsen, A pullback attractor for an asymptotically autonomous multivalued Cauchy problem with spatially variable exponent, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[16]

P. E. Kloeden and Me ihua Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[17]

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

[18]

C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165–198. doi: 10.1016/0022-247X(82)90160-3.  Google Scholar

[19]

K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001) 59–78. Google Scholar

[20]

M. Růžička, Flow of shear dependent elecrorheological fluids, C. R. Acad. Sci. Paris, Série I, 329 (1999), 393-398.  doi: 10.1016/S0764-4442(00)88612-7.  Google Scholar

[21]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[22]

J. Simsen and J. Valero, Characterization of Pullback Attractors for Multivalued Nonautonomous Dynamical Systems, Advances in Dynamical Systems and Control, 179–195, Stud. Syst. Decis. Control, 69, Springer, [Cham], 2016.  Google Scholar

[23]

J. Simsen and E. Capelato, Some properties for exact generalized processes, Continuous and Distributed Systems II, 209–219, Studies in Systems, Decision and Control. 1ed. 30, Springer International Publishing, 2015. doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[24]

J. Simsen and C. B. Gentile, On p-Laplacian differential inclusions-Global existence, compactness properties and asymptotic behavior, Nonlinear Analysis, 71 (2009), 3488-3500.  doi: 10.1016/j.na.2009.02.044.  Google Scholar

[25]

J. Simsen and M. S. Simsen, Existence and upper semicontinuity of global attractors for $p(x)$-Laplacian systems, J. Math. Anal. Appl., 388 (2012), 23–38. doi: 10.1016/j.jmaa.2011.10.003.  Google Scholar

[26]

J. Simsen and M. S. Simsen, On asymptotically autonomous dynamics for multivalued evolution problems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), no. 8, 3557–3567. doi: 10.3934/dcdsb.2018278.  Google Scholar

[27]

J. Simsen and P. Wittbold, Compactness results with applications for nonautonomous coupled inclusions, J. Math. Anal. Appl., 479 (2019), 426–449., doi: 10.1016/j.jmaa.2019.06.033.  Google Scholar

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[29]

I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Editon, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995.  Google Scholar

show all references

References:
[1]

C. O. Alves, S. Shmarev, J. Simsen and M. S. Simsen, The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: existence and asymptotic behavior, J. Math. Anal. Appl., 443 (2016), 265-294.  doi: 10.1016/j.jmaa.2016.05.024.  Google Scholar

[2]

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

[3]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Berlin, 1990.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[8]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[9]

P. Harjulehto, P. Hästö, U. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Analysis, 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.  Google Scholar

[10]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[11]

P. E. Kloeden and P. Marín-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, J. Setvalued & Variational Analysis, 19 (2011), 43-57.  doi: 10.1007/s11228-009-0123-2.  Google Scholar

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc. Providence, 2011. doi: 10.1090/surv/176.  Google Scholar

[13]

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

[14]

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

[15]

P. E. Kloeden, J. Simsen and M. S. Simsen, A pullback attractor for an asymptotically autonomous multivalued Cauchy problem with spatially variable exponent, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[16]

P. E. Kloeden and Me ihua Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[17]

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

[18]

C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165–198. doi: 10.1016/0022-247X(82)90160-3.  Google Scholar

[19]

K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001) 59–78. Google Scholar

[20]

M. Růžička, Flow of shear dependent elecrorheological fluids, C. R. Acad. Sci. Paris, Série I, 329 (1999), 393-398.  doi: 10.1016/S0764-4442(00)88612-7.  Google Scholar

[21]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[22]

J. Simsen and J. Valero, Characterization of Pullback Attractors for Multivalued Nonautonomous Dynamical Systems, Advances in Dynamical Systems and Control, 179–195, Stud. Syst. Decis. Control, 69, Springer, [Cham], 2016.  Google Scholar

[23]

J. Simsen and E. Capelato, Some properties for exact generalized processes, Continuous and Distributed Systems II, 209–219, Studies in Systems, Decision and Control. 1ed. 30, Springer International Publishing, 2015. doi: 10.1007/978-3-319-19075-4_12.  Google Scholar

[24]

J. Simsen and C. B. Gentile, On p-Laplacian differential inclusions-Global existence, compactness properties and asymptotic behavior, Nonlinear Analysis, 71 (2009), 3488-3500.  doi: 10.1016/j.na.2009.02.044.  Google Scholar

[25]

J. Simsen and M. S. Simsen, Existence and upper semicontinuity of global attractors for $p(x)$-Laplacian systems, J. Math. Anal. Appl., 388 (2012), 23–38. doi: 10.1016/j.jmaa.2011.10.003.  Google Scholar

[26]

J. Simsen and M. S. Simsen, On asymptotically autonomous dynamics for multivalued evolution problems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), no. 8, 3557–3567. doi: 10.3934/dcdsb.2018278.  Google Scholar

[27]

J. Simsen and P. Wittbold, Compactness results with applications for nonautonomous coupled inclusions, J. Math. Anal. Appl., 479 (2019), 426–449., doi: 10.1016/j.jmaa.2019.06.033.  Google Scholar

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[29]

I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Editon, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995.  Google Scholar

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