Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis 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 (2006) 551-597] and continued in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, AN Carvalho. G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz. Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are Upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in L(p) and H(1) norms. (C) 2008 Elsevier Inc. All rights reserved.

MEC

MEC[PHB2006-003-PC]

MEC[MTM2006-08262]

MEC

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

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

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

CNPq[305447/2005-0]

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]

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

FAPESP[2008/53094-4]

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

FAPESP[06/04781-3]

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

FAPESP[07100981-0]