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The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem
Asymptotic behavior of coupled inclusions with variable exponents
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 |
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.
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. |
[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. |
[3] |
J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Berlin, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc. Providence, 2011.
doi: 10.1090/surv/176. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001) 59–78. |
[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. |
[21] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[29] |
I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Editon, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995. |
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. |
[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. |
[3] |
J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Berlin, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc. Providence, 2011.
doi: 10.1090/surv/176. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn., 13 (2001) 59–78. |
[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. |
[21] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[29] |
I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Editon, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995. |
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