Asymptotic Analysis of a Phase-Field Model with Memory

A phase-field model accounting for memory effects is considered. This model consists of a hyperbolic integrodifferential equation coupled with a parabolic differential inclusion. The latter relation rules the evolution of the phase field and contains a time relaxation parameter which happens to be very small in the appli- cations. A well-posed initial and boundary value problem for the evolution system is introduced and the asymptotic behavior of its solution as the time relaxation goes to zero is analyzed rigorously. Convergence results and error estimates are obtained under suitable assumptions ensuring that the limit problem has a unique solution.