Metastability for scalar conservation laws in a bounded domain

The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval $I=(-\ell,\ell)$ is considered, with emphasis on the metastable dynamics, whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coecient $\epsilon>0$ goes to zero. We describe such behavior by deriving an ODE for the position of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions $\{U^\epsilon(\cdot;\xi)\}$ , parametrized by the location of the shock layer $\xi$, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at $U^\epsilon$, we estimate the size of the layer location.