Arrieta Algarra, José MaríaCarvalho, Alexandre N.Lozada-Cruz, Germán2023-06-202023-06-2020090022-039610.1016/j.jde.2008.12.014https://hdl.handle.net/20.500.14352/42008In this paper we conclude the analysis started in [3] and continued in [4] con- cerning the behavior of the asymptotic dynamics of a dissipative reactions di_usion equation in a dumbbell domain as the channel shrinks to a line segment. In [3], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [4], 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 normsengjournal articlehttp://www.sciencedirect.com/science/journal/00220396open access517.9Hyperbolic equilibriaShrinking channelsLower semicontinuousEcuaciones diferenciales1202.07 Ecuaciones en Diferencias