Global existence and decay for solutions of the Hele-Shaw flow with injection

We examine the stability and decay of the free boundary perturbations in a Hele-Shaw cell under the injection of fluid. In particular, we study the perturbations of spherical boundaries as time t → +∞. In the presence of positive surface tension, we examine both slow and fast injection rates.When fluid is injected slowly, the perturbations decay back to an expanding sphere exponentially fast, while for fast injection, the perturbation decays to an expanding sphere with an algebraic rate. In the absence of surface tension, we study the case of a constant injection rate, and prove that perturbations of the sphere decay like (1 + t)^-1/2+∈ for ∈ > 0 small.