@article{Lo2023, 
author = {Catharine W. K. Lo and José Francisco Rodrigues},
title = {On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {3},
pages = {1-38},
keywords = {Stefan problem, fractional derivatives, boundary value problem, nonlocal diffusion, phase transitions, subdifferential, nonlinear, fractional evolution equation},
url = {https://www.sciopen.com/article/10.3934/mine.2023047},
doi = {10.3934/mine.2023047},
abstract = {In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain    Ω  ⊂            R        d   with time-dependent Dirichlet boundary condition for the temperature    ϑ  =  ϑ  (  x  ,  t  ),    ϑ  =  g on        Ω    c    ×  ]  0  ,  T  [, and initial condition        η    0   for the enthalpy    η  =  η  (  x  ,  t  ), given in    Ω  ×  ]  0  ,  T  [ by               ∂      η              ∂      t        +            L        A    s    ϑ  =  f    with  η  ∈  β  (  ϑ  )  ,where              L        A    s   is an anisotropic fractional operator defined in the distributional sense by     ⟨            L        A    s    u  ,  v  ⟩  =      ∫                            R                d              A      D    s    u  ⋅      D    s    v    d  x  ,   β 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  ↗  1 towards the classical local problem, the asymptotic behaviour as    t  →  ∞, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph    β.}
}