Large-time behaviour of Hele-Shaw flow with injection or suction for perturbed balls in $R^N$

Large time behaviour of Hele–Shaw flow with surface tension and with injection or suction in one point is discussed. We consider domains that are initially small perturbations of balls. Radially, symmetric solutions are stationary after a suitable time-dependent rescaling. The evolution of perturbations can be described by a non-local non-linear parabolic evolution equation. Global existence results and decay properties are derived using energy estimates in Sobolev spaces.