GAKUTO International Series Mathematical Sciences and Applications, Vol.23(2005) Mathematical Approach to Nonlinear Phenomena; Modelling, Analysis and Simulations, pp.55–72 THE REACTION–DIFFUSION PROBLEM ON DUMBBELL DOMAINS Maria Gokieli and Nicolas Varchon Interdisciplinary Centre for Mathematical and Computational Modelling, Warsaw University Zwirki i Wigury 93, 02-089 Warsaw, Poland E-mail: mgokieli@icm.edu.pl, varchon@icm.edu.pl Abstract. We consider a reaction-diffusion equation defined on a nonconvex domain Ω n consisting of a disconnected set Ω and a shrinking connection between its components. We study the flux properties and prove that each stable (unstable) equilibrium on Ω is approached by a unique sequence of stable (unstable) equilibria on Ω n . AMS Subject Classification 37C25, 35B40 c Gakk¯ otosho 200?, GAKK ¯ OTOSHO CO.,LTD