# American Institute of Mathematical Sciences

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

 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.

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:

show all references

##### References:
 [1] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 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, 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, 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, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901 [8] Jiapeng Huang, Chunhua Jin. Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5415-5439. doi: 10.3934/dcds.2020233 [9] 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 [10] Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054 [11] 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 [12] 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 [13] Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301 [14] 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 [15] Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246 [16] Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382 [17] 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 [18] 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 [19] 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 [20] 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

2020 Impact Factor: 1.081