The fractional Stefan problem: global regularity of the bounded selfsimilar solution

Abstract

Abstract 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 $$h(x,t)\ge 0$$ h ( x , t ) ≥ 0 , the evolution equation reads $$ \partial _t h + (-\Delta )^s \Phi (h) = 0 \qquad \text {in } \mathbb {R} \times (0,T), $$ ∂ t h + ( - Δ ) s Φ ( h ) = 0 in R × ( 0 , T ) , where $$u = \Phi (h) := (h-L)_+ = \max \{h-L,0\}$$ u = Φ ( h ) : = ( h - L ) + = max { h - L , 0 } denotes the temperature, $$L>0$$ L > 0 is the latent heat, and the fractional parameter $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . We prove that the regularity of the self-similar solution depends on s , with a critical threshold at $$s = 1/2$$ s = 1 / 2 . More precisely, in the subcritical case $$0< s < 1/2$$ 0 < s < 1 / 2 , the self-similar solution exhibits at least $$C^{1,\alpha }$$ C 1 , α regularity in all of $$\mathbb {R}$$ R , with Hölder exponent $$\alpha >0$$ α >