On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion

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March 2017, 6(1): 1-13. doi: 10.3934/eect.2017001

On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion

María Astudillo and Marcelo M. Cavalcanti ^,

Department of Mathematics, State University of Maringá, 87020-900 Maringá PR, Brazil

* Corresponding author: Marcelo M. Cavalcanti

Received March 2016 Revised August 2016 Published December 2016

In this article, we are concerned with the asymptotic behavior of a class of degenerate parabolic problems involving porous medium type equations, in a bounded domain, when the diffusion coefficient becomes large. We prove the upper semicontinuity of the associated global attractor as the diffusion increases to infinity.

Keywords: Porous medium, large diffusion, attractors, reaction diffusion equation.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35K55, 35K57.

Citation: María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

References:

[1]	H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.
[2]	F. Andreu, J. M. Mazón, F. Simondon and J. Toledo, Attractor for a degenerate nonlinear diffusion problem with nonlinear boundary condition, J. Dynam. Differential Equations, 10 (1998), 347-377. doi: 10.1023/A:1022640912144.
[3]	J. M. Arrieta, A. N. Carvalho, N. Alexandre and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59. doi: 10.1006/jdeq.2000.3876.
[4]	A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q.
[5]	E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.
[6]	A. Eden, B. Michaux and J. M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dynam. Differential Equations, 3 (1991), 87-131. doi: 10.1007/BF01049490.
[7]	I. Ekland and R. Temam, Analyse Convexe et Problémes Variationnels Dunod, Paris, 1976.
[8]	M. Efendiev and S. Zelik, Finite and infinite-dimensional attractors for porous media equations, Proc. Lond. Math. Soc., 96 (2008), 51-77. doi: 10.1112/plms/pdm026.
[9]	E. Feireisl, Ph. Laurençot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Differential Equations, 129 (1996), 239-261. doi: 10.1006/jdeq.1996.0117.
[10]	M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Bio. Sci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.
[11]	J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8.
[12]	N. Igbida, A nonlinear diffusion problem with localized large diffusion, Comm. Partial Differential Equations, 29 (2004), 647-670. doi: 10.1081/PDE-120037328.
[13]	N. Igbida and F. Karami, Some competition phenomena in evolution equations, Adv. Math. Sci. Appl., 17 (2007), 559-587.
[14]	O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Lezioni Lincee, 1991. doi: 10.1017/CBO9780511569418.
[15]	A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Appl. Math., 57 (2012), 43-69. doi: 10.1007/s10492-012-0004-0.
[16]	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.

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References:

[1]	H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.
[2]	F. Andreu, J. M. Mazón, F. Simondon and J. Toledo, Attractor for a degenerate nonlinear diffusion problem with nonlinear boundary condition, J. Dynam. Differential Equations, 10 (1998), 347-377. doi: 10.1023/A:1022640912144.
[3]	J. M. Arrieta, A. N. Carvalho, N. Alexandre and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59. doi: 10.1006/jdeq.2000.3876.
[4]	A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q.
[5]	E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.
[6]	A. Eden, B. Michaux and J. M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dynam. Differential Equations, 3 (1991), 87-131. doi: 10.1007/BF01049490.
[7]	I. Ekland and R. Temam, Analyse Convexe et Problémes Variationnels Dunod, Paris, 1976.
[8]	M. Efendiev and S. Zelik, Finite and infinite-dimensional attractors for porous media equations, Proc. Lond. Math. Soc., 96 (2008), 51-77. doi: 10.1112/plms/pdm026.
[9]	E. Feireisl, Ph. Laurençot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Differential Equations, 129 (1996), 239-261. doi: 10.1006/jdeq.1996.0117.
[10]	M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Bio. Sci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.
[11]	J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8.
[12]	N. Igbida, A nonlinear diffusion problem with localized large diffusion, Comm. Partial Differential Equations, 29 (2004), 647-670. doi: 10.1081/PDE-120037328.
[13]	N. Igbida and F. Karami, Some competition phenomena in evolution equations, Adv. Math. Sci. Appl., 17 (2007), 559-587.
[14]	O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Lezioni Lincee, 1991. doi: 10.1017/CBO9780511569418.
[15]	A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Appl. Math., 57 (2012), 43-69. doi: 10.1007/s10492-012-0004-0.
[16]	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.

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