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An asymptotic analysis for a system with equation and dynamic boundarycondition of Cahn-Hilliard type is carried out as the coefficient of thesurface diffusion acting on the phase variable tends to 0, thus obtaining aforward-backward dynamic boundary condition at the limit. This is done in avery general setting, with nonlinear terms admitting maximal monotone graphsboth in the bulk and on the boundary. The two graphs are related by a growthcondition, with the boundary graph that dominates the other one. It turns outthat in the limiting procedure the solution of the problem looses someregularity and the limit equation has to be properly interpreted in the senseof a subdifferential inclusion. However, the limit problem is still well-posedsince a continuous dependence estimate can be proved. Moreover, in the casewhen the two graphs exhibit the same growth, it is shown that the solutionenjoys more regularity and the boundary condition holds almost everywhere. Anerror estimate can also be shown, for a suitable order of the diffusionparameter.

The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity