  In this paper we study the long time dynamics of the solutions to the
initial-boundary value problem for a scalar conservation law with a saturating
nonlinear diffusion. After discussing the existence of a unique stationary
solution and its asymptotic stability, we focus our attention on the phenomenon
of 'metastability', whereby the time-dependent solution develops into a layered
function in a relatively short time, and subsequently approaches a steady state
in a very long time interval. Numerical simulations illustrate the results.
