Curved thin domains and parabolic equations

M. Prizzi ; M. Rinaldi ; K. P. Rybakowski

Studia Mathematica, Tome 151 (2002), p. 109-140 / Harvested from The Polish Digital Mathematics Library

Résumé

Consider the family uₜ = Δu + G(u), t > 0, $x ∈ Ω_{ε}$, $∂_{ν_{ε}}u = 0$, t > 0, $x ∈ ∂Ω_{ε}$, $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$. If G is dissipative, then equation $(E_{ε})$ has a global attractor $_{ε}$. We identify a “limit” equation for the family $(E_{ε})$, prove convergence of trajectories and establish an upper semicontinuity result for the family $_{ε}$ as ε → 0⁺.

Détail

Publié le : 2002-01-01

Zbl 0999.35005

EUDML-ID : urn:eudml:doc:286445