Grain Boundary Grooving in a Bicrystal with Passivation Coating

We use the sixth order linear parabolic equation \begin{equation} \frac{\partial y}{\partial t}=B\left(\alpha\frac{\partial^{6}y}{\partial x^{6}}-\frac{\partial^{4}y}{\partial x^{4}}\right),\ x\in \mathbb{R}_{+},\ t>0,\nonumber \end{equation} proposed by Rabkin and describing the evolution of a solid surface covered with a thin, inert and fully elastic passivation layer, to analyze the grain boundary groove formation on initially flat surface. We derive the corresponding boundary conditions and construct an asymptotic representation of the solution to this initial boundary value problem when $\alpha$ is small, by applying the theory of singular perturbation. We illustrate the effect of passivation film near and far from a grain boundary groove.