|| This talk addresses a large-time rescaling behavior of Hele-Shaw cells for large data initial domains. The Polubarinova-Galin equation is the reformulation of zero surface tension Hele-Shaw ows with injection at the origin in two dimensions by considering the moving domain $\Omega(t) = f(B_1(0),t)$ for some Riemann mapping $f(\xi,t)$. We give a sharp
large-time rescaling behavior of global strong polynomial solutions to this equation and hence obtain a sharp rescaling behavior of the moving domains. Furthermore, we also show that a small perturbation of the initial function of a global strong polynomial solution also gives rise to a global strong solution and a large-time rescaling behavior of the moving domain is shown as well.