Curvature-driven surface diffusion on crystalline surfaces is modeled by  the nonlinear 4th order Mullins equation. There is a class of  weakly anisotropic nonlinear models that is fully integrable. Exact solutions are constructed for development of a grain boundary groove and for smoothing of an initial ramp dislocation.  For the grooving problem, a piecewise nonlinear but solvable  model can be made arbitrarily close to the isotropic nonlinear Mullins equation. The solution shows that unlike in the linear model, the groove depth remains bounded  as the dihedral angle approaches vertical.

At a dislocation point of infinite curvature,  the quasilinear Mullins model should be extended to a fully nonlinear   model to account for Gibbs-Thompson evaporation-condensation. An exactly solvable fully nonlinear degenerate diffusion model  shows that unlike in the quasilinear model, deposition rate at the dislocation point is bounded, and the slope remains discontinuous for a finite time.

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