On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

Abstract

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 \[\frac{\partial η}{\partial t} +\mathcal{L}_A^s \vartheta= f\quad\text{ with }η\in β(\vartheta),\] where ${\mathcal{L}}_A^s$ is an anisotropic fractional operator defined in the distributional sense by \[\langle\mathcal{L}_A^su,v\rangle=\int_{\mathbb{R}^d}AD^su\cdot D^sv\,dx,\] $\beta$ 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 $\beta$.

Final version, to appear in Mathematics in Engineering

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