Metastable dynamics and spatially inhomogeneous equilibria in dumbbell-shaped domains

The motion of internal layers for three singularly perturbed reaction diffusion problems, including the Allen-Cahn equation, is studied in a two-dimensional dumbbell-shaped domain. The channel region that connects the two attachments, or lobes, of the dumbbell is taken to be rectangular. The motion of straight-line internal layers in the channel region is analyzed by using an asymptotic projection method. It is shown that this motion is metastable and highly dependent on the local convexity properties of the boundary near the contact region between the ends of the channel and the two attachments. When the domain is nonconvex it is shown that the metastable internal layers dynamics in the channel tends, as t --> infinity, to a limiting, stable, spatially inhomogeneous equilibrium solution.