Internal Layers, Small Eigenvalues, and the Sensitivity of Metastable Motion

On a semi-infinite domain, an analytical characterization of exponentially slow internal layer motion for the Allen–Cahn equation and for a singularly perturbed viscous shock problem is given. The results extend some previous results that were restricted to a finite geometry. For these slow motion problems, we show that the slow dynamics associated with the semi-infinite domain are not preserved, even qualitatively, by imposing a commonly used form of artificial boundary condition to truncate the semi-infinite domain to a finite domain. This extreme sensitivity to boundary conditions and domain truncation is a direct result of the exponential ill-conditioning of the underlying linearized problem. For Burgers equation, many of the analytical results are verified by calculating certain explicit solutions. Some related ill-conditioned internal layer problems are examined.