Dynamics in dumbbell domains II. The limiting problem

In this work we continue the analysis of the asymptotic dynamics of reaction-diffusion problems in a dumbbell domain started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2) (2006) 551-597]. Here we study the limiting problem, that is, an evolution problem in a ""domain"" which consists of an open, bounded and smooth set Omega subset of R(N) with a curve R(0) attached to it. The evolution in both parts of the domain is governed by a parabolic equation. In Omega the evolution is independent of the evolution in R(0) whereas in R(0) the evolution depends on the evolution in Omega through the continuity condition of the solution at the junction points. We analyze in detail the linear elliptic and parabolic problem, the generation of linear and nonlinear semigroups, the existence and structure of attractors. (C) 2009 Elsevier Inc. All rights reserved.

MEC[PHB2006-003-PC]

MEC

MEC[MTM2006-08262]

MEC

Programa de Financiacion de Grupos de Investigacion UCM-Comunidad de Madrid, Spain

Programa de Financiacion de Grupos de Investigacion UCM-Comunidad de Madrid, Spain[CCG07-UCM/ESP-2393]

Programa de Financiacion de Grupos de Investigacion UCM-Comunidad de Madrid, Spain[920894]

Programa de Financiacion de Grupos de Investigacion UCM-Comunidad de Madrid, Spain

SIMUMAT-Comunidad de Madrid, Spain

SIMUMAT-Comunidad de Madrid, Spain

CNPq[305447/2005-0]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

CNPq[451761/2008-1]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

DGU

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

CAPES/DGU[267/2008]

FAPESP, Brazil[2008/53094-4]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

FAPESP, Brazil[06/04781-3]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

FAPESP, Brazil[07/00981-0]