  This note is concerned with a nonlinear diffusion problem of phase-field
type, consisting of a parabolic system of two partial differential equations,
complemented by boundary and initial conditions. The system arises from a model
of two-species phase segregation on an atomic lattice and was introduced by
Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The two unknowns are the
phase parameter and the chemical potential. In contrast to previous
investigations about this PDE system, we consider here a dynamic boundary
condition for the phase variable that involves the Laplace-Beltrami operator
and models an additional nonconserving phase transition occurring on the
surface of the domain. We are interested to some asymptotic analysis and first
discuss the asymptotic limit of the system as the viscosity coefficient of the
order parameter equation tends to 0: the convergence of solutions to the
corresponding solutions for the limit problem is proven. Then, we study the
long-time behavior of the system for both problems, with positive or zero
viscosity coefficient, and characterize the omega-limit set in both cases.
