  An asymptotic analysis for a system with equation and dynamic boundary
condition of Cahn-Hilliard type is carried out as the coefficient of the
surface diffusion acting on the phase variable tends to 0, thus obtaining a
forward-backward dynamic boundary condition at the limit. This is done in a
very general setting, with nonlinear terms admitting maximal monotone graphs
both in the bulk and on the boundary. The two graphs are related by a growth
condition, with the boundary graph that dominates the other one. It turns out
that in the limiting procedure the solution of the problem looses some
regularity and the limit equation has to be properly interpreted in the sense
of a subdifferential inclusion. However, the limit problem is still well-posed
since a continuous dependence estimate can be proved. Moreover, in the case
when the two graphs exhibit the same growth, it is shown that the solution
enjoys more regularity and the boundary condition holds almost everywhere. An
error estimate can also be shown, for a suitable order of the diffusion
parameter.
