Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, and K. P. Rybakowski. "Curved thin domains and parabolic equations." Studia Mathematica 151.2 (2002): 109-140. <http://eudml.org/doc/286445>.

@article{M2002,
abstract = {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⁺.},
author = {M. Prizzi, M. Rinaldi, K. P. Rybakowski},
journal = {Studia Mathematica},
keywords = {curved squeezed domains; limit problem; semilinear Neumann boundary value problems; upper semicontinuity result},
language = {eng},
number = {2},
pages = {109-140},
title = {Curved thin domains and parabolic equations},
url = {http://eudml.org/doc/286445},
volume = {151},
year = {2002},
}

TY - JOUR
AU - M. Prizzi
AU - M. Rinaldi
AU - K. P. Rybakowski
TI - Curved thin domains and parabolic equations
JO - Studia Mathematica
PY - 2002
VL - 151
IS - 2
SP - 109
EP - 140
AB - 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⁺.
LA - eng
KW - curved squeezed domains; limit problem; semilinear Neumann boundary value problems; upper semicontinuity result
UR - http://eudml.org/doc/286445
ER -