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We study the global existence and decay to spherical equilibrium of Hele-Shawflows with surface tension. We prove that without injection of fluid,perturbations of the sphere decay to zero exponentially fast. On the otherhand, with a time-dependent rate of fluid injection into the Hele-Shaw cell,the distance from the moving boundary to an expanding sphere (withtime-dependent radius) also decays to zero but with an algebraic rate, whichdepends on the injection rate of the fluid

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