Well-posedness of the weak formulation for the phase-field model with memory

Abstract

A phase-field model based on the Gurtin-Pipkin heat flux law is considered. This model consists in a Volterra integrodifferential equation of hyperbolic type coupled with a nonlinear parabolic equation. The system is then associated with a set of initial and Neumann boundary conditions. The resulting problem was already studied by the authors who proved existence and uniqueness of a smooth solution. A~careful and detailed investigation on weak solutions is the goal of this paper, going from the aspects of the approximation to the proof of continuous dependence estimates. In addition, a sufficient condition for the boundedness of the phase variable is given.