The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution"

We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads $$\{\begin{array}{ll}{\mathrm{\partial }}_{t}h+(-\mathrm{\Delta }{)}^s\mathrm{\Phi }(h)=0& \text{in }{\mathbb{R}}^n\times (0,T),\\ h(\cdot ,0)=h_0& \text{in }{\mathbb{R}}^n,\end{array}$$ where $u=\mathrm{\Phi }(h):=(h-L{)}_{+}=\max \{h-L,0\}$ denotes the temperature, $L>0$ is the latent heat, and $s\in (0,1)$. We prove that the regularity of the self-similar solution depends on $s$, with a critical threshold at $s=1\mathrm{/}2$. More precisely, in the subcritical case $0<s<1\mathrm{/}2$, the self-similar solution exhibits at least $C^{1,\alpha }$ regularity, with Hölder exponent $\alpha >0$. In contrast, we show that the enthalpy of the self-similar solution is not Lipschitz continuous at the free boundary in the critical case $s=1\mathrm{/}2$, as well as in the supercritical case $1\mathrm{/}2<s<1$. Additional results are also established concerning the lateral regularity at the free boundary and the asymptotic behavior of the solution profile as $x\to \pm \mathrm{\infty }$.