This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.

中文翻译：

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具有不连续扩散系数的退化非线性抛物线方程

本文致力于研究一些具有不连续扩散强度的非线性抛物线方程。此类问题自然会出现在物理和生物模型中。我们的分析基于变分技术，特别是概率度量空间中的梯度流，该空间配备了 Monge-Kantorovich 最优传输问题中出现的距离。相关的内能泛函一般不能微分，因此经典结果不能直接应用于我们的设置。我们研究了线性和多孔介质类型扩散的组合，并在合适的 Sobolev 空间中的分布意义上展示了解的存在性和唯一性。我们的解决方案概念使我们能够对新兴的关键区域进行精细的表征，先前在数值实验中观察到。还指出了与三相自由边界问题的联系。