American Institute of Mathematical Sciences

Advanced
December  2004, 3(4): 637-651. doi: 10.3934/cpaa.2004.3.637

Spatial homogeneity in parabolic problems with nonlinear boundary conditions

Alexandre Nolasco de Carvalho 1, and  Marcos Roberto Teixeira Primo 2,

1. 

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil
2. 

Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil

Received  October 2003 Revised  May 2004 Published  September 2004

In this work we prove that global attractors of systems of weakly coupled parabolic equations with nonlinear boundary conditions and large diffusivity are close to attractors of an ordinary differential equation. The limiting ordinary differential equation is given explicitly in terms of the reaction, boundary flux, the $n$-dimensional Lebesgue measure of the domain and the $(n-1)-$Hausdorff measure of its boundary. The tools are invariant manifold theory and comparison results.
Keywords: comparison and positivity results, spatial homogeneity, Invariant manifold theory, global attractors.
Mathematics Subject Classification: 34D35, 34D45, 35B05, 35B40, 35K4.
Citation: Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637
[1]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079
[2]

Alexey G. Mazko. Positivity, robust stability and comparison of dynamic systems. Conference Publications, 2011, 2011 (Special) : 1042-1051. doi: 10.3934/proc.2011.2011.1042
[3]

Bernold Fiedler, Carlos Rocha. Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5099-5122. doi: 10.3934/dcds.2014.34.5099
[4]

Arrigo Cellina, Carlo Mariconda, Giulia Treu. Comparison results without strict convexity. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 57-65. doi: 10.3934/dcdsb.2009.11.57
[5]

George Osipenko. Linearization near a locally nonunique invariant manifold. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 189-205. doi: 10.3934/dcds.1997.3.189
[6]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305
[7]

Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221
[8]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[9]

Julián López-Gómez, Paul H. Rabinowitz. The effects of spatial heterogeneities on some multiplicity results. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 941-952. doi: 10.3934/dcds.2016.36.941
[10]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[11]

Michał Jóźwikowski, Witold Respondek. A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems. Journal of Geometric Mechanics, 2019, 11 (1) : 77-122. doi: 10.3934/jgm.2019005
[12]

Vittorino Pata. Two questions arising in the theory of attractors. Evolution Equations and Control Theory, 2019, 8 (3) : 663-668. doi: 10.3934/eect.2019031
[13]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013
[14]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[15]

Monica Conti, Vittorino Pata. On the regularity of global attractors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1209-1217. doi: 10.3934/dcds.2009.25.1209
[16]

Miguel Ângelo De Sousa Mendes. Quasi-invariant attractors of piecewise isometric systems. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 323-338. doi: 10.3934/dcds.2003.9.323
[17]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[18]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[19]

T.K. Subrahmonian Moothathu. Homogeneity of surjective cellular automata. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 195-202. doi: 10.3934/dcds.2005.13.195
[20]

Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga. Determining the global manifold structure of a continuous-time heterodimensional cycle. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022008

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy