  We prove the existence of Ricci flow starting from a class of metrics with
unbounded curvature, which are doubly-warped products over an interval with a
spherical factor pinched off at an end. These provide a forward evolution from
some known and conjectured finite-time local singularities of Ricci flow,
generalizing previous examples. The class also includes metrics with
non-compact singular ends which become instantaneously compact. Furthermore, we
prove local stability of the forward evolution, which allows us to glue it to
other manifolds and create a forward evolution from spaces which are not
globally warped products.
