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 of R-d with time-dependent Dirichlet boundary condition for the temperature theta = theta(x, t), theta = g on Omega(c)x]0, T[, and initial condition eta(0) for the enthalpy eta = eta(x, t), given in Omega x]0, T[ by inverted iota eta/partial derivative t L-A(s)theta = f with eta is an element of beta(theta), where L-A(s) is an anisotropic fractional operator defined in the distributional sense by < L(A)(s)u, v > = integral(Rd) AD(s)u . 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 NE arrow 1 towards the classical local problem, the asymptotic behaviour as t -> infinity, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph beta.