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November  2016, 21(9): 3239-3267. doi: 10.3934/dcdsb.2016096

Global attractors for $p$-Laplacian differential inclusions in unbounded domains

Jacson Simsen 1, and  José Valero 2,

1. 

Instituto de Matemática e Computaçã, Universidade Federal de Itajubá, 37500-903 Itajubá MG
2. 

Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202

Received  October 2015 Revised  March 2016 Published  October 2016

In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.
Keywords: upper semicontinuity., differential inclusions, Unbounded domains, $p$-Laplacian operator, attractors.
Mathematics Subject Classification: 35B40, 35B41, 35K55, 37L.
Citation: Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096
References:
[1]

C. T. Anh and T. D. Ke, Long-time behavior for quasilinear parabolic equations involving weighted $p$-Laplacian operators, Nonlinear Anal., 71 (2009), 4415-4422. doi: 10.1016/j.na.2009.02.125.  Google Scholar

[2]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023.  Google Scholar

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J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, Boston, 2009.  Google Scholar

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A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498.  Google Scholar

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[7]

S. M. Bruschi, C. B. Gentile and M. R. T. Primo, Continuity properties on $p$ for $p$-Laplacian parabolic problems, Nonlinear Anal., 72 (2010), 1580-1588. doi: 10.1016/j.na.2009.08.044.  Google Scholar

[8]

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H. Brézis, Operateurs Maximaux Monotones et Semi-groups de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

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T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., Series S, 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17.  Google Scholar

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V. L. Carbone, C. B. Gentile and K. Schiabel-Silva, Asymptotic properties in parabolic problems dominated by a $p$-Laplacian operator with localized large diffusion, Nonlinear Anal., 74 (2011), 4002-4011. doi: 10.1016/j.na.2011.03.028.  Google Scholar

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A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators, Bolletino U.M.I., 2 (1999), 693-706.  Google Scholar

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A. N. Carvalho and C. B. Gentile, Asymptotic behaviour of non-linear parabolic equations with monotone principal part, J. Math. Anal. Appl., 280 (2003), 252-272. doi: 10.1016/S0022-247X(03)00037-4.  Google Scholar

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M. A. Efendiev and M. Ôtani, Infinite-dimensional attractors for parabolic equations with $p$-Laplacian in heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 565-582. doi: 10.1016/j.anihpc.2011.03.006.  Google Scholar

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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.  Google Scholar

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N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004.  Google Scholar

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N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusions in unbounded domain, in Continuous and distributed systems, Solid Mechanics and its Applications, 211 (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), Springer International Publishing, Switzerland, 2014, 205-220. doi: 10.1007/978-3-319-03146-0_15.  Google Scholar

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A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615. doi: 10.1016/j.jmaa.2005.05.003.  Google Scholar

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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 Contin. Dyn. Syst., 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.  Google Scholar

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O. V. Kapustyan, O. P. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion system without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906. doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

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O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of the uniform global attractor for general non-autonomous reaction-diffusion systems, in Continuous and distributed systems, Solid Mechanics and its Applications, 211 (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), Springer International Publishing Switzerland, 2014, 163-180. doi: 10.1007/978-3-319-03146-0_12.  Google Scholar

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O. V. Kapustyan and V. S. Melnik, On global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.  Google Scholar

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O. V. Kapustyan and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191.  Google Scholar

[31]

P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernet. Systems Anal., 47 (2011), 800-811. doi: 10.1007/s10559-011-9359-6.  Google Scholar

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P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218. doi: 10.1134/S0001434612070231.  Google Scholar

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P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8.  Google Scholar

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show all references

References:
[1]

C. T. Anh and T. D. Ke, Long-time behavior for quasilinear parabolic equations involving weighted $p$-Laplacian operators, Nonlinear Anal., 71 (2009), 4415-4422. doi: 10.1016/j.na.2009.02.125.  Google Scholar

[2]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023.  Google Scholar

[3]

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

[4]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, Boston, 2009.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498.  Google Scholar

[6]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.  Google Scholar

[7]

S. M. Bruschi, C. B. Gentile and M. R. T. Primo, Continuity properties on $p$ for $p$-Laplacian parabolic problems, Nonlinear Anal., 72 (2010), 1580-1588. doi: 10.1016/j.na.2009.08.044.  Google Scholar

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[9]

H. Brézis, Operateurs Maximaux Monotones et Semi-groups de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[10]

T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., Series S, 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17.  Google Scholar

[11]

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

[12]

V. L. Carbone, C. B. Gentile and K. Schiabel-Silva, Asymptotic properties in parabolic problems dominated by a $p$-Laplacian operator with localized large diffusion, Nonlinear Anal., 74 (2011), 4002-4011. doi: 10.1016/j.na.2011.03.028.  Google Scholar

[13]

A. N. Carvalho and T. Dlotko, Partly dissipative systems in uniformly local spaces, Colloquium Mathematicum, 100 (2004), 221-242. doi: 10.4064/cm100-2-6.  Google Scholar

[14]

A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators, Bolletino U.M.I., 2 (1999), 693-706.  Google Scholar

[15]

A. N. Carvalho and C. B. Gentile, Asymptotic behaviour of non-linear parabolic equations with monotone principal part, J. Math. Anal. Appl., 280 (2003), 252-272. doi: 10.1016/S0022-247X(03)00037-4.  Google Scholar

[16]

M. A. Efendiev and M. Ôtani, Infinite-dimensional attractors for parabolic equations with $p$-Laplacian in heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 565-582. doi: 10.1016/j.anihpc.2011.03.006.  Google Scholar

[17]

E. Feireisl, P. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $\mathbbR^n$, C. R. Acad. Sci. Paris. Sér. I Math., 319 (1994), 147-151.  Google Scholar

[18]

E. Feireisl, P. Laurencot and F. Simondon, Compact attractors for degenerate parabolic equations in $\mathbbR^n$, C. R. Acad. Sci. Paris. Sér. I Math., 320 (1995), 1079-1083.  Google Scholar

[19]

E. Feireisl and J. Norbury, Some existence and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh, 119 (1991), 1-17. doi: 10.1017/S0308210500028262.  Google Scholar

[20]

C. B. Gentile and M. R. T. Primo, Parameter dependent quasi-linear parabolic equations, Nonlinear Anal., 59 (2004), 801-812. doi: 10.1016/S0362-546X(04)00292-5.  Google Scholar

[21]

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.  Google Scholar

[22]

N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004.  Google Scholar

[23]

N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusions in unbounded domain, in Continuous and distributed systems, Solid Mechanics and its Applications, 211 (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), Springer International Publishing, Switzerland, 2014, 205-220. doi: 10.1007/978-3-319-03146-0_15.  Google Scholar

[24]

A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonlinear Anal., 71 (2009), 155-171. doi: 10.1016/j.na.2008.10.037.  Google Scholar

[25]

A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615. doi: 10.1016/j.jmaa.2005.05.003.  Google Scholar

[26]

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 Contin. Dyn. Syst., 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[27]

O. V. Kapustyan, O. P. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion system without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906. doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[28]

O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of the uniform global attractor for general non-autonomous reaction-diffusion systems, in Continuous and distributed systems, Solid Mechanics and its Applications, 211 (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), Springer International Publishing Switzerland, 2014, 163-180. doi: 10.1007/978-3-319-03146-0_12.  Google Scholar

[29]

O. V. Kapustyan and V. S. Melnik, On global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.  Google Scholar

[30]

O. V. Kapustyan and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191.  Google Scholar

[31]

P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernet. Systems Anal., 47 (2011), 800-811. doi: 10.1007/s10559-011-9359-6.  Google Scholar

[32]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218. doi: 10.1134/S0001434612070231.  Google Scholar

[33]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8.  Google Scholar

[34]

P. E. Kloeden and J. Valero, Attractors of weakly asymptotically compact setvalued dynamical systems, Set-Valued Anal., 13 (2005), 381-404. doi: 10.1007/s11228-004-0047-9.  Google Scholar

[35]

S. Lian, W. Gao, C. Cao and H. Yuan, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342 (2008), 27-38. doi: 10.1016/j.jmaa.2007.11.046.  Google Scholar

[36]

J. L. Lions, Quelques Methodes de Resolutions Des Problemes Aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[37]

S. Ma and H. Li, Global attractors for weighted $p$-Laplacian equations with boundary degeneracy, J. Math. Phys., 53 (2012), 012701, 8 pp. doi: 10.1063/1.3675441.  Google Scholar

[38]

N. Mastorakis and H. Fathabadi, On the solution of p-Laplacian for non-Newtonian fluid flow, WSEAS Trans. Math., 8 (2009), 238-245.  Google Scholar

[39]

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

[40]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbbR^N$ with continuous nonlinearity, Asymptotic Analysis, 44 (2005), 111-130.  Google Scholar

[41]

W. Niu and C. Zhong, Global attractors for the $p$-Laplacian equations with nonregular data, J. Math. Anal. Appl., 392 (2012), 123-135. doi: 10.1016/j.jmaa.2012.03.025.  Google Scholar

[42]

H. El. Ouardi, Global attractor for quasilinear parabolic systems involving weighted $p$-Laplacian operators, J. Pure Appl. Math. Adv. Appl., 5 (2011), 79-97.  Google Scholar

[43]

M. Ôtani, On existence of strong solutions for $\frac{du}{dt}+\partial\psi^{1}(u( t)) -\partial\psi^{2}( u( t\))$ ∋ $f(t) $, J. Fac. Sci. Univ. Tokio. Sect. IA Math., 24 (1977), 575-605.  Google Scholar

[44]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, J. Differential Equations, 46 (1982), 286-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar

[45]

A. Rodríguez-Bernal and B. Wang, Attractors for partly dissipative reaction diffusion systems in $\mathbbR^N$, J. Math. Anal. Appl., 252 (2000), 790-803. doi: 10.1006/jmaa.2000.7122.  Google Scholar

[46]

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