Asymptotic behaviour of injection and suction for Hele-Shaw flow in R with surface tension near balls

We discuss long-time behaviour of the Hele-Shaw flow in R with surface tension and injection or suction in the origin, for domains that are small perturbations of balls. After rescaling, radially symmetric solutions become stationary. We study the stability of these solutions. In particular, we show that all liquid can be removed by suction if the suction point and the geometric centre coincide and the ratio of suction speed and surface tension is small enough. Any smaller amount of liquid can be removed if the suction point is near the geometric centre. We use the principle of linearised stability and the abstract theory of quasilinear parabolic equations.