Metastable dynamics and spatially inhomogeneous equilibria in dumbbell-shaped domains

The motion of internal layers for three singularly perturbed reaction diffu sion problems, including the Allen-Cahn equation, is studied in a two-dimen sional dumbbell-shaped domain. The channel region that connects the two att achments, 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 metastabl e and highly dependent on the local convexity properties of the boundary ne ar the contact region between the ends of the channel and the two attachmen ts. When the domain is nonconvex it is shown that the metastable internal l ayers dynamics in the channel tends, as t --> infinity, to a limiting, stab le, spatially inhomogeneous equilibrium solution.