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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. 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. doi: 10.1016/j.na.2003.09.023. Google Scholar

[3]

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

[4]

J. P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkhäusser, (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. 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. 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. doi: 10.1016/j.na.2009.08.044. Google Scholar

[8]

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

[9]

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

[10]

T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations,, Discrete Contin. Dyn. Syst., 2 (2009), 17. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1017/S0308210500028262. Google Scholar

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C. B. Gentile and M. R. T. Primo, Parameter dependent quasi-linear parabolic equations,, Nonlinear Anal., 59 (2004), 801. 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. 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. 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, (2014), 205. 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. 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. 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. 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. 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, (2014), 163. 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. 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. 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. 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. 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. 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. 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. 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, (1969). Google Scholar

[37]

S. Ma and H. Li, Global attractors for weighted $p$-Laplacian equations with boundary degeneracy,, J. Math. Phys., 53 (2012). 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. Google Scholar

[39]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. 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. 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. 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. 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. Google Scholar

[44]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators,, J. Differential Equations, 46 (1982), 286. 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. doi: 10.1006/jmaa.2000.7122. Google Scholar

[46]

J. Simsen, A note on $p$-Laplacian parabolic problems in $\mathbbR^n$,, Nonlinear Anal., 75 (2012), 6620. doi: 10.1016/j.na.2012.08.007. Google Scholar

[47]

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

[48]

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

[49]

J. Simsen and C. B. Gentile, Well-posed $p$-Laplacian problems with large diffusion,, Nonlinear Anal., 71 (2009), 4609. doi: 10.1016/j.na.2009.03.041. Google Scholar

[50]

J. Simsen and C. B. Gentile, Systems of $p$-Laplacian differential inclusions with large diffusion,, J. Math. Anal. Appl., 368 (2010), 525. doi: 10.1016/j.jmaa.2010.02.006. Google Scholar

[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. Google Scholar

[52]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[53]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation,, SIAM J. Math. Anal., 14 (1983), 1107. doi: 10.1137/0514086. Google Scholar

[54]

D. Terman, A free boundary arising from a model for nerve conduction,, J. Differential Equations, 58 (1985), 345. doi: 10.1016/0022-0396(85)90004-X. Google Scholar

[55]

A. A. Tolstonogov, On solutions of evolution inclusions.I,, Sibirsk. Mat. Zh., 33 (1992), 161. doi: 10.1007/BF00970899. Google Scholar

[56]

H. Wang and C. Chen, On global attractors for $(p,q)$-Laplacian parabolic system in $\mathbbR^N,$, Indagationes Mathematicae, 23 (2012), 423. doi: 10.1016/j.indag.2012.02.007. Google Scholar

[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. doi: 10.1016/j.na.2005.11.004. 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. 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. doi: 10.1016/j.na.2003.09.023. Google Scholar

[3]

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

[4]

J. P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkhäusser, (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. 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. 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. doi: 10.1016/j.na.2009.08.044. Google Scholar

[8]

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

[9]

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

[10]

T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations,, Discrete Contin. Dyn. Syst., 2 (2009), 17. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2014), 205. 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. 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. 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. 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. 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, (2014), 163. 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. 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. 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. 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. 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. 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. 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. 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, (1969). Google Scholar

[37]

S. Ma and H. Li, Global attractors for weighted $p$-Laplacian equations with boundary degeneracy,, J. Math. Phys., 53 (2012). 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. Google Scholar

[39]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. 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. 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. 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. 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. Google Scholar

[44]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators,, J. Differential Equations, 46 (1982), 286. 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. doi: 10.1006/jmaa.2000.7122. Google Scholar

[46]

J. Simsen, A note on $p$-Laplacian parabolic problems in $\mathbbR^n$,, Nonlinear Anal., 75 (2012), 6620. doi: 10.1016/j.na.2012.08.007. Google Scholar

[47]

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

[48]

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

[49]

J. Simsen and C. B. Gentile, Well-posed $p$-Laplacian problems with large diffusion,, Nonlinear Anal., 71 (2009), 4609. doi: 10.1016/j.na.2009.03.041. Google Scholar

[50]

J. Simsen and C. B. Gentile, Systems of $p$-Laplacian differential inclusions with large diffusion,, J. Math. Anal. Appl., 368 (2010), 525. doi: 10.1016/j.jmaa.2010.02.006. Google Scholar

[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. Google Scholar

[52]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[53]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation,, SIAM J. Math. Anal., 14 (1983), 1107. doi: 10.1137/0514086. Google Scholar

[54]

D. Terman, A free boundary arising from a model for nerve conduction,, J. Differential Equations, 58 (1985), 345. doi: 10.1016/0022-0396(85)90004-X. Google Scholar

[55]

A. A. Tolstonogov, On solutions of evolution inclusions.I,, Sibirsk. Mat. Zh., 33 (1992), 161. doi: 10.1007/BF00970899. Google Scholar

[56]

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