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

中文翻译：

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多孔介质方程两相扩展的大时间行为

我们研究了多孔介质方程的两相扩展解的长时间行为，该方程模拟了所谓的海水入侵问题。目标是确定与问题的重新缩放版本的稳定状态相对应的自相似解决方案。我们完全表征了独特的稳态，这些稳态被确定为凸能量的最小值并显示为径向对称。此外，我们证明了时间相关模型的解随着时间趋于无穷大而收敛于唯一的静止状态。我们最终提供了静止状态的数值说明，并展示了数值收敛率。