American Institute of Mathematical Sciences

Advanced
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. Igbida, A nonlinear diffusion problem with localized large diffusion, Comm. Partial Differential Equations, 29 (2004), 647-670.  doi: 10.1081/PDE-120037328.  Google Scholar

[13]

N. Igbida and F. Karami, Some competition phenomena in evolution equations, Adv. Math. Sci. Appl., 17 (2007), 559-587.   Google Scholar

[14]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Lezioni Lincee, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. Igbida, A nonlinear diffusion problem with localized large diffusion, Comm. Partial Differential Equations, 29 (2004), 647-670.  doi: 10.1081/PDE-120037328.  Google Scholar

[13]

N. Igbida and F. Karami, Some competition phenomena in evolution equations, Adv. Math. Sci. Appl., 17 (2007), 559-587.   Google Scholar

[14]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Lezioni Lincee, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[2]

Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks & Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669
[3]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[4]

Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649
[5]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495
[6]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[7]

Youshan Tao, Michael Winkler. Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901
[8]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[9]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114
[10]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
[11]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101
[12]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[13]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[14]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[15]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[16]

Yang Wang, Yi-fu Feng. $ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 461-481. doi: 10.3934/naco.2019027
[17]

Antonio Carlos Fernandes, Marcela Carvalho Gonçcalves, Jacson Simsen. Non-autonomous reaction-diffusion equations with variable exponents and large diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1485-1510. doi: 10.3934/dcdsb.2018217
[18]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[19]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[20]

Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (30)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences