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On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion
Department of Mathematics, State University of Maringá, 87020-900 Maringá PR, Brazil |
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.
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. Google Scholar |
[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. |
show all references
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. Google Scholar |
[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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