Justification of a hyperbolic approach to phase changes in materials with memory

Abstract

This paper is concerned with phase transition problems in materials with memory. Two different sets of constitutive assumptions are considered for the internal energy and the heat flux, one leading to a parabolic integro-differential equation and the other giving rise to a hyperbolic one. Both of them are coupled with phase change laws and suitable initial and boundary conditions. While the first class of resulting problems corresponds to a well-established and widely investigated approach, the second model could be partly arguable from a constitutive point of view. Here, it is shown that actually the hyperbolic phase change problems are nothing but singular perturbations of the parabolic ones. By performing a rigorous asymptotic analysis, we prove convergence results and deduce error estimates.