The one-phase fractional Stefan problem
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We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in R^N. The equation for the enthalpy h reads ∂_t h+ (-Δ)^s Φ(h) =0 where the temperature u:=Φ(h):=max{h-L,0} is defined for some constant L>0 called the latent heat, and (-Δ)^s is the fractional Laplacian with exponent s∈(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)=ξ_0 t^1/(2s) for some ξ_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→0^+, L→ +∞, s→0^+ and s→ 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.
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