SPATIAL HOMOGENEITY IN PARABOLIC PROBLEMS WITH
Transcrição
SPATIAL HOMOGENEITY IN PARABOLIC PROBLEMS WITH
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 3, Number 4, December 2004 Website: http://AIMsciences.org pp. 637–651 SPATIAL HOMOGENEITY IN PARABOLIC PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS Alexandre Nolasco de Carvalho Departamento de Matemática Instituto de Ciências Matemáticas e de Computação Caixa postal 668, 13560-970 São Carlos, São Paulo, Brasil Marcos Roberto Teixeira Primo Departamento de Matemática, Universidade Estadual de Maringá 87020-900 Maringá, Paraná, Brasil (Communicated by Alain Miranville) 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. 1. Introduction. Let Ω ⊂ RN , N ∈ N, be a bounded domain with smooth boundary Γ := ∂Ω. Consider the following problem ut (t, x) = Du(t, x) + F (u(t, x)) t > 0, x ∈ Ω, (1.1) D ∂u (t, x) = G(u(t, x)) t > 0, x ∈ Γ, ∂ν where u = (u1 , u2 , · · · , un ) , n ≥ 1, ∂u = (∇u1 , ν , · · · ∇un , ν ) , ν is the out∂ν ward normal vector and D is the matrix ⎡ ⎤ d1 0 0 · · · 0 ⎢ 0 d2 0 · · · 0 ⎥ ⎢ ⎥ ⎢ ⎥ D := ⎢ 0 0 d3 · · · 0 ⎥ (1.2) ⎢ .. ⎥ .. .. . . . . ⎣ . ⎦ . . . . 0 0 0 · · · dn n×n with di > 0, i = 1, · · · , n. The nonlinearities F = (F1 , · · · , Fn ), G = (G1 , · · · , Gn ) : Rn →Rn are locally Lipschitz functions. Throughout the paper we assume, without loss of generality, that |Ω| = 1. 2000 Mathematics Subject Classification. 34D35, 34D45, 35B05, 35B40, 35K40. Key words and phrases. Invariant manifold theory, global attractors, comparison and positivity results, spatial homogeneity. 637
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