Metastable Dynamics and Exponential Asymptotics in Multi-dimensional Domains

Certain singularly perturbed partial diierential equations exhibit a phenomenon known as dynamic metastability, whereby the solution evolves on an asymptotically exponentially long time interval as the singular perturbation parameter tends to zero. This article illustrates a technique to analyze metastable behavior for a range of problems in multi-dimensional domains. The problems considered include the exit problem for diiusion in a potential well, models of interface propagation in materials science, an activator-inhibitor model in mathematical biology, and a ame-front problem. Many of these problems can be formulated in terms of non-local partial diierential equations. This non-local feature is shown to be essential to the existence of metastable behavior.