Curved thin domains and parabolic equations

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

doi:10.4064/sm151-2-2

Curved thin domains and parabolic equations

M. Prizzi
, M. Rinaldi
, K. P. Rybakowski

Research output: Contribution to journal › Article › peer-review

Abstract

Consider the family u_t = Δu + G(u), t > 0, x ∈ Ω_ε (E_ε) ∂_νε u = 0, t > O, x ∈ ∂ Ω_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_epsi;) has a global attractor A_ε. We identify a "limit" equation for the family (E_ε), prove convergence of trajectories and establish an upper semicontinuity result for the family A_ε as ε → 0⁺.

Original language	English
Pages (from-to)	109-140
Number of pages	32
Journal	Studia Mathematica
Volume	151
Issue number	2
DOIs	https://doi.org/10.4064/sm151-2-2
Publication status	Published - 2002
Externally published	Yes

Keywords

Attractors
Curved squeezed domains
Reaction-diffusion equations

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10.4064/sm151-2-2

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pages = "109--140",

journal = "Studia Mathematica",

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T1 - Curved thin domains and parabolic equations

AU - Prizzi, M.

AU - Rinaldi, M.

AU - Rybakowski, K. P.

PY - 2002

Y1 - 2002

N2 - Consider the family ut = Δu + G(u), t > 0, x ∈ Ωε (Eε) ∂νε u = 0, t > O, x ∈ ∂ Ωε of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ωε is a thin domain in ℝl, possibly with holes, which collapses, as ε → 0+, onto a (curved) k-dimensional submanifold of ℝl. If G is dissipative, then equation (Eepsi;) has a global attractor Aε. We identify a "limit" equation for the family (Eε), prove convergence of trajectories and establish an upper semicontinuity result for the family Aε as ε → 0+.

AB - Consider the family ut = Δu + G(u), t > 0, x ∈ Ωε (Eε) ∂νε u = 0, t > O, x ∈ ∂ Ωε of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ωε is a thin domain in ℝl, possibly with holes, which collapses, as ε → 0+, onto a (curved) k-dimensional submanifold of ℝl. If G is dissipative, then equation (Eepsi;) has a global attractor Aε. We identify a "limit" equation for the family (Eε), prove convergence of trajectories and establish an upper semicontinuity result for the family Aε as ε → 0+.

KW - Attractors

KW - Curved squeezed domains

KW - Reaction-diffusion equations

UR - https://www.scopus.com/pages/publications/0036324731

U2 - 10.4064/sm151-2-2

DO - 10.4064/sm151-2-2

M3 - Article

SN - 0039-3223

VL - 151

SP - 109

EP - 140

JO - Studia Mathematica

JF - Studia Mathematica

IS - 2

ER -

Curved thin domains and parabolic equations

Abstract

Keywords

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