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Dynamics in dumbbell domains III. Continuity of attractors |
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| Authors: | José M Arrieta Alexandre N Carvalho German Lozada-Cruz |
| Institution: | a Dep. Matemática Aplicada, Fac. Matemáticas, Univ. Complutense de Madrid, 28040 Madrid, Spain b Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, CP 668, 13560-970 São Carlos SP, Brazil c Departamento de Matemática, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000 São José do Rio Preto SP, Brazil |
| Abstract: | In this paper we conclude the analysis started in J.M. Arrieta, A.N. 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, A.N. 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, A.N. 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, A.N. 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 Lp and H1 norms. |
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