  It is shown that a hypersurface of a space form is the initial data for a
solution to the mean curvature flow by parallel hypersurfaces if, and only if,
it is isoparametric. By solving an ordinary differential equation, explicit
solutions are given for all isoparametric hypersurfaces of space forms. In
particular, for such hypersurfaces of the sphere, the exact collapsing time
into a focal submanifold is given in terms of its dimension, the principal
curvatures and their multiplicities.
