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We study the large time behavior of the solutions to a two phase extension ofthe porous medium equation, which models the so-called seawater intrusionproblem. The goal is to identify the self-similar solutions that correspond tosteady states of a rescaled version of the problem. We fully characterize theunique steady states that are identified as minimizers of a convex energy andshown to be radially symmetric. Moreover, we prove the convergence of thesolution to the time-dependent model towards the unique stationary state astime goes to infinity. We finally provide numerical illustrations of thestationary states and we exhibit numerical convergence rates.

Large time behavior of a two phase extension of the porous medium equation