The one-phase fractional Stefan problem

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. The equation for the enthalpy $h$ reads $\partial_t h+ (-\Delta)^s \Phi(h) =0$ where the temperature $u:=\Phi(h):=\max\{h-L,0\}$ is defined for some constant $L>0$ called the latent heat, and $(-\Delta)^s$ is the fractional Laplacian with exponent $s\in(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$ located at $x(t)=\xi_0 t^{1/(2s)}$ for some $\xi_0>0$. We also provide well-posedness and basic properties of very weak solutions for general bounded data $h_0$. The temperatures $u$ 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 $u$ for solutions with compactly supported initial temperatures. We also show the property of conservation of positivity for $u$ so that the support never recedes. On the contrary, the enthalpy $h$ has infinite speed of propagation and we obtain precise estimates on the tail. The limits $L\to0^+$, $L\to +\infty$, $s\to0^+$ and $s\to 1^-$ 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.