Dynamics in dumbbell domains. II: The limiting problem

Abstract

In this work we continue the analysis of the asymptotic dynamics of reaction–diffusion problems in a dumbbell domain 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 (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 Ω⊂RN with a curve R0 attached to it. The evolution in both parts of the domain is governed by a parabolic equation. In Ω the evolution is independent of the evolution in R0 whereas in R0 the evolution depends on the evolution in Ω 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.