Communications on Pure and Applied Analysis (CPAA)

Spatial homogeneity in parabolic problems with nonlinear boundary conditions

Pages: 637 - 651, Volume 3, Issue 4, December 2004      doi:10.3934/cpaa.2004.3.637

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Alexandre Nolasco de Carvalho - Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil (email)
Marcos Roberto Teixeira Primo - Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil (email)

Abstract: 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:  Invariant manifold theory, global attractors, comparison and positivity results, spatial homogeneity
Mathematics Subject Classification:  34D35, 34D45, 35B05, 35B40, 35K40.

Received: October 2003;      Revised: May 2004;      Published: September 2004.