Global solutions to bubble growth in porous media

We study a moving boundary problem modeling an injected fluid into another viscous fluid. The viscous fluid is withdrawn at infinity and governed by Darcyʼs law. We present solutions to the free boundary problem in terms of time-derivative of a generalized Newtonian potentials of the characteristic function of the bubble. This enables us to show that the bubble occupies the entire space as the time tends to infinity if and only if the internal generalized Newtonian potential of the initial bubble is a quadratic polynomial. Howison (1985) [7], and DiBenedetto and Friedman (1986) [2], studied such behavior, but for bounded bubbles. We extend their results to unbounded bubbles.