Abstract
We prove the existence of a one-parameter family of solutions of the porous medium equation in which the interface is a half line whose end point advances at a constant speed. Then we prove the stability of the solutions under a suitable class of perturbations. We discuss the relevance of these solutions to gravity-driven flows of thin films, and show that some solutions develop a very thin triangular plateau in the direction of propagation and that the angle of the plateau and its thickness are decreasing functions of the speed.
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