An asymptotic analysis for a nonstandard cahn-hilliard system with viscosity

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter p and the chemical potential ?; each equation includes a viscosity term { respectively, ??t? and ? ?tp { with ? and ? two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its (? ?)solutions. Here we discuss the asymptotic limit of the system as " tends to 0. We prove convergence of (? ?)solutions to the corresponding solutions for the case ? = 0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.