We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.
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Joachim Escher, Philippe Laurençot, Bogdan-Vasile Matioc, Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 4, pp. 583–598DOI 10.1016/J.ANIHPC.2011.04.001