Metastability and Pinning for Convection-Di usion-Reaction Equations in Thin Domains

Two singularly perturbed convection-diiusion-reaction equations are examined to show the eeect of small spatial inhomogeneities on metastable dynamics in one spatial dimension. The two problems that are considered are the Ginzburg-Landau equation from phase separation theory and a viscous shock problem modeling transonic nozzle ow. For each problem, the diierential operator is perturbed by an exponentially small spatially inhomogeneous term as the singular perturbation parameter " tends to zero. This weak spatially inhomogeneous term represents the perturbing eeect on the metastable dynamics of an internal layer that is slowly propagating along a channel of slowly varying cross-sectional area. It is shown that the eeect of such a perturbation can be very signiicant and often leads to the existence of new stable equilibrium internal layer solutions that do not exist in the absence of the perturbation. This pinning eeect induced by the perturbation is studied asymptotically as " ! 0 and the asymptotic results are compared with full numerical results.