On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta =\vartheta (x,t)$, $\vartheta =g$ on ${\mathrm{\Omega }}^c\times ]0,T[$, and initial condition ${\eta }_0$ for the enthalpy $\eta =\eta (x,t)$, given in $\mathrm{\Omega }\times ]0,T[$ by

$$\frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}+{\mathcal{L}}_A^s\vartheta =f\text{with}\eta \in \beta (\vartheta ),$$

where ${\mathcal{L}}_A^s$ is an anisotropic fractional operator defined in the distributional sense by

$$⟨{\mathcal{L}}_A^{s}u,v⟩={\int }_{{\mathbb{R}}^d}A{D}^{s}u\cdot D^{s}vdx,$$

$\beta$ is a maximal monotone graph, $A(x)$ is a symmetric, strictly elliptic and uniformly bounded matrix, and $D^s$ is the distributional Riesz fractional gradient for $0<s<1$. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $s\nearrow 1$ towards the classical local problem, the asymptotic behaviour as $t\to \mathrm{\infty }$, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $\beta$.