  We study the evolution of the "Momentarily Static and Radiation Free" (MSRF)
initial data for the Apostolatos - Thorne cylindrical shell model. We analyze
the relation between the parameters characterizing the MSRF data those for the
corresponding final static configuration, and show that there is a priori no
conflict for any choice of initial MSRF data, in contrast with some recent
results of Nakao, Ida and Kurita. We also consider the problem in the linear
approximation, and show that the evolution is stable in all cases. We find that
the approach to the final state is very slow, with an inverse logarithmic
dependence on time at fixed radius. To complement these results we introduce a
numerical computation procedure that allows us to visualize the explicit form
of the evolution of the shell and of the gravitational field up to large times.
The results are in agreement with the qualitative behaviour conjectured by
Apostolatos and Thorne, with an initial damped oscillatory stage, but with
oscillations about a position that approaches slowly that of the static final
state, as indicated by our analysis. We also include an Appendix, where we
prove the existence of solutions of the cylindrical wave equation with
vanishing initial value for $r > R_0$, ($R_0 > 0$ some finite constant), that
approach a constant value for large times. This result is crucial for the proof
of compatibility of arbitrary MSRF initial data and a final static
configuration for the system.
