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A system with equation and dynamic boundary condition of Cahn-Hilliard typeis considered. This system comes from a derivation performed in Liu-Wu (Arch.Ration. Mech. Anal. 233 (2019), 167--247) via an energetic variationalapproach. Actually, the related problem can be seen as a transmission problemfor the phase variable in the bulk and the corresponding variable on theboundary. The asymptotic behavior as the coefficient of the surface diffusionacting on the boundary phase variable goes to 0 is investigated. By thisanalysis 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 thelimit 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 moreregular and we prove an error estimate for a suitable order of the diffusionparameter.

A Cahn-Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials