  In this work, we give a perturbation theorem for strong polynomial solutions
to the zero surface tension Hele-Shaw equation driven by injection or suction,
so called the Polubarinova-Galin equation. This theorem enables us to explore
properties of solutions with initial functions close to but are not polynomial.
Applications of this theorem are given in the suction or injection case. In the
former case, we show that if the initial domain is close to a disk, most of
fluid will be sucked before the strong solution blows up. In the later case, we
obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows
in terms of invariant Richardson complex moments. This rescaling behavior
result generalizes a recent result regarding large-time rescaling behavior for
small data in terms of moments. As a byproduct of a theorem in this paper, a
short proof of existence and uniqueness of strong solutions to the
Polubarinova-Galin equation is given.
