We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] called the latent heat, and [Formula: see text] stands for the fractional Laplacian with exponent [Formula: see text].We prove the existence of a continuous and bounded selfsimilar solution of the form [Formula: see text] which exhibits a free boundary at the change-of-phase level [Formula: see text]. This level is located at the line (called the free boundary) [Formula: see text] for some [Formula: see text]. The construction is done in 1D, and its extension to [Formula: see text]-dimensional space is shown.We also provide well-posedness and basic properties of very weak solutions for general bounded data [Formula: see text] in several dimensions. The temperatures [Formula: see text] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [Formula: see text] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [Formula: see text] so that the support never recedes. On the contrary, the enthalpy [Formula: see text] has infinite speed of propagation and we obtain precise estimates on the tail.The limits [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.

中文翻译：

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单相分数 Stefan 问题

我们研究[公式：见正文]中提出的分数扩散单相 Stefan 问题的解的存在性和性质以及自由边界。就焓[公式：见文本]而言，演化方程为[公式：见文本]，而温度被定义为[公式：见文本]对于一些称为潜热的常数[公式：见文本]，以及[公式：见正文]代表带指数的分数拉普拉斯算子[公式：见正文]。我们证明了形式[公式：见正文]的连续有界自相似解的存在，它在变化时表现出自由边界-阶段级别[公式：见正文]。此级别位于线（称为自由边界）[公式：参见文本] 处为某些 [公式：参见文本]。构造是在一维中完成的，并且它扩展到 [公式：见文本]维空间。我们还为一般有界数据[公式：见文本]在几个维度上提供了非常弱解的适定性和基本属性。这些解的温度 [公式：见正文] 是具有有限传播速度的连续函数，具有可能的自由边界。对于具有紧凑支持的初始温度的解决方案，我们获得了[公式：参见文本]支持时间增长的估计。此外，我们展示了[公式：见正文]的正性守恒性质，因此支持永远不会退去。相反，焓[公式：见文]具有无限的传播速度，我们在尾部得到精确估计。极限[公式：见文]、[公式：见文]、[公式：见文]和[公式：见正文]也有探讨，我们发现了与经过充分研究的扩散问题的有趣联系。最后，我们提出了收敛单调有限差分方案，并包括旨在说明一些获得的理论结果以及其他有趣现象的数值实验。