  A system with equation and dynamic boundary condition of Cahn-Hilliard type
is considered. This system comes from a derivation performed in Liu-Wu (Arch.
Ration. Mech. Anal. 233 (2019), 167--247) via an energetic variational
approach. Actually, the related problem can be seen as a transmission problem
for the phase variable in the bulk and the corresponding variable on the
boundary. The asymptotic behavior as the coefficient of the surface diffusion
acting on the boundary phase variable goes to 0 is investigated. By this
analysis we obtain a forward-backward dynamic boundary condition at the limit.
We can deal with a general class of potentials having a double-well structure,
including the non-smooth double-obstacle potential. We illustrate that the
limit problem is well-posed by also proving a continuous dependence estimate.
Moreover, in the case when the two graphs, in the bulk and on the boundary,
exhibit the same growth, we show that the solution of the limit problem is more
regular and we prove an error estimate for a suitable order of the diffusion
parameter.
