Dynamics in dumbbell domains III. Continuity of attractors

Autor(es)

Arrieta, Jose M. ^[1] ; Carvalho, Alexandre N. ^[2] ; Lozada-Cruz, German ^[3]

Afiliação do(s) autor(es)

^[1] Univ Complutense Madrid, Dept Matemat Aplicada, Fac Matemat, E-28040 Madrid - Spain

^[2] Univ Sao Paulo, Dept Matemat, Inst Ciencias Matemat & Computacao, BR-13560970 Sao Carlos, SP - Brazil

^[3] Univ Estadual Paulista, Dept Matemat, IBILCE, UNESP, BR-15054000 Sao Jose Dos Campos - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Journal of Differential Equations; v. 247, n. 1, p. 225-259, JUL 1 2009.
• Citações Web of Science: 22

Resumo

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. (AU)

• Processo FAPESP: 08/53094-4 - Sistemas dinâmicos em espaços de dimensão infinita sob perturbações
• Beneficiário: Alexandre Nolasco de Carvalho
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 06/04781-3 - Dinâmica em domínios tipo Dumbbell: continuidade de atratores
• Beneficiário: German Jesus Lozada Cruz
• Modalidade de apoio: Auxílio à Pesquisa - Regular