Global attractors for p-Laplacian differential inclusions in unbounded domains

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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

Abstract
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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 and 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.
[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.
[3]	J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[4]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, Boston, 2009.
[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.
[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.
[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.
[8]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.
[9]	H. Brézis, Operateurs Maximaux Monotones et Semi-groups de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[36]	J. L. Lions, Quelques Methodes de Resolutions Des Problemes Aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.
[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.
[38]	N. Mastorakis and H. Fathabadi, On the solution of p-Laplacian for non-Newtonian fluid flow, WSEAS Trans. Math., 8 (2009), 238-245.
[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.
[40]	F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbbR^N$ with continuous nonlinearity, Asymptotic Analysis, 44 (2005), 111-130.
[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.
[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.
[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.
[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.
[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.
[46]	J. Simsen, A note on $p$-Laplacian parabolic problems in $\mathbbR^n$, Nonlinear Anal., 75 (2012), 6620-6624. doi: 10.1016/j.na.2012.08.007.
[47]	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.
[48]	J. Simsen and C. B. Gentile, On $p$-Laplacian differential inclusions-global existence, compactness properties and asymptotic behavior, Nonlinear Anal., 71 (2009), 3488-3500. doi: 10.1016/j.na.2009.02.044.
[49]	J. Simsen and C. B. Gentile, Well-posed $p$-Laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617. doi: 10.1016/j.na.2009.03.041.
[50]	J. Simsen and C. B. Gentile, Systems of $p$-Laplacian differential inclusions with large diffusion, J. Math. Anal. Appl., 368 (2010), 525-537. doi: 10.1016/j.jmaa.2010.02.006.
[51]	J. Simsen and E. N. Neres Junior, Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion, Bulletin of Parana's Mathematical Society, 33 (2015), 235-245.
[52]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[53]	D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129. doi: 10.1137/0514086.
[54]	D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363. doi: 10.1016/0022-0396(85)90004-X.
[55]	A. A. Tolstonogov, On solutions of evolution inclusions.I, Sibirsk. Mat. Zh., 33 (1992), 161-174. doi: 10.1007/BF00970899.
[56]	H. Wang and C. Chen, On global attractors for $(p,q)$-Laplacian parabolic system in $\mathbbR^N,$ Indagationes Mathematicae, 23 (2012), 423-437. doi: 10.1016/j.indag.2012.02.007.
[57]	M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $\mathbbR^n$, Nonlinear Anal., 66 (2007), 1-13. doi: 10.1016/j.na.2005.11.004.
[58]	M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142. doi: 10.1016/j.jmaa.2006.04.085.
[59]	C. Zhong and W. Niu, On the $Z_{2}$ index of the global attractor for a class of $p$-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704. doi: 10.1016/j.na.2010.07.022.
[60]	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, Springer Proc. Math. Stat., 95, Springer, Cham, 2015, pp.283-294. doi: 10.1007/978-3-319-08377-3_28.
[61]	M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and J. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Long-time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Series: Advances in Mechanics and Mathematics, Vol. 27, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28512-7.

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.
[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.
[3]	J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[4]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, Boston, 2009.
[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.
[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.
[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.
[8]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.
[9]	H. Brézis, Operateurs Maximaux Monotones et Semi-groups de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[36]	J. L. Lions, Quelques Methodes de Resolutions Des Problemes Aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.
[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.
[38]	N. Mastorakis and H. Fathabadi, On the solution of p-Laplacian for non-Newtonian fluid flow, WSEAS Trans. Math., 8 (2009), 238-245.
[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.
[40]	F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbbR^N$ with continuous nonlinearity, Asymptotic Analysis, 44 (2005), 111-130.
[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.
[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.
[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.
[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.
[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.
[46]	J. Simsen, A note on $p$-Laplacian parabolic problems in $\mathbbR^n$, Nonlinear Anal., 75 (2012), 6620-6624. doi: 10.1016/j.na.2012.08.007.
[47]	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.
[48]	J. Simsen and C. B. Gentile, On $p$-Laplacian differential inclusions-global existence, compactness properties and asymptotic behavior, Nonlinear Anal., 71 (2009), 3488-3500. doi: 10.1016/j.na.2009.02.044.
[49]	J. Simsen and C. B. Gentile, Well-posed $p$-Laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617. doi: 10.1016/j.na.2009.03.041.
[50]	J. Simsen and C. B. Gentile, Systems of $p$-Laplacian differential inclusions with large diffusion, J. Math. Anal. Appl., 368 (2010), 525-537. doi: 10.1016/j.jmaa.2010.02.006.
[51]	J. Simsen and E. N. Neres Junior, Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion, Bulletin of Parana's Mathematical Society, 33 (2015), 235-245.
[52]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.
[53]	D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129. doi: 10.1137/0514086.
[54]	D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363. doi: 10.1016/0022-0396(85)90004-X.
[55]	A. A. Tolstonogov, On solutions of evolution inclusions.I, Sibirsk. Mat. Zh., 33 (1992), 161-174. doi: 10.1007/BF00970899.
[56]	H. Wang and C. Chen, On global attractors for $(p,q)$-Laplacian parabolic system in $\mathbbR^N,$ Indagationes Mathematicae, 23 (2012), 423-437. doi: 10.1016/j.indag.2012.02.007.
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