Hyperbolic Phase Change Problems in Heat Conduction with Memory

摘要

The aim of the paper is to formulate and study phase transition problems inmaterials with memory, based on the Gurtin--Pipkin constitutive assumption on the heat flux. As different phases are involved, the internal energy is allowed to depend on the phase variable (besides the temperature) and to take its past history into account. By considering the standard equilibrium condition at the interface between two phases, the authors deal with a hyperbolic Stefan problem reckoning with memory effects. Then, substituting the equilibrium condition with a relaxation dynamics, the authors represent some dissipation phenomena including supercooling or superheating. The application of a fixed point argument helps us to show the existence and uniqueness of the solution to the latter problem (still of hyperbolic type). Hence, by introducing a suitable regularization and taking the limit as a kinetic parameter goes to zero, the authors prove an existence result for the former Stefan problem. Moreover, its uniqueness is deduced by contradiction.