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 $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by

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

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

\begin{document}$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $\end{document}

$ \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\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 $.