  The parabolic N-membranes problem for the p-Laplacian and the complete order
constraint on the components of the solution is studied in what concerns the
approximation, the regularity and the stability of the variational solutions.
We extend to the evolutionary case the characterization of the Lagrange
multipliers associated with the ordering constraint in terms of the
characteristic functions of the coincidence sets. We give continuous dependence
results, and study the asymptotic behavior as $t \to \infty$ of the solution
and the coincidence sets, showing that they converge to their stationary
counterparts.
