Large-time rescaling behaviors of some rational type solutions to the Polubarinova-Galin equation with injection

The main goal of this paper is to give a precise description of rescaling behaviors of rational type global strong solutions to the Polubarinova-Galin equation. The Polubarinova-Galin equation is the reformulation of the zero surface tension Hele-Shaw problem with a single source at the origin by considering the moving domain as the Riemann mapping of the unit disk centered at the origin. The coefficients ${a_{k}(t)}_{kgeq 2}$ of the polynomial strong solution $f_{k_{0}}(xi,t)=sum_{i=1}^{k_{0}}a_{i}(t)xi^{i}$ decay to zero algebraically as $t^{-lambda_{k}}$ ($lambda_{k}=k/2$) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.