On self-similar solutions to the surface diffusion flow equations with contact angle boundary conditions

Abstract

We consider the surface diffusion flow equation when the curve is given as the graph of a function $v(x,t)$ defined in a half line ${\mathbf{R}}^{+}=\{x>0\}$ under the boundary conditions $v_x=\tan \beta >0$ and $v_{xxx}=0$ at $x=0$. We construct a unique (spatially bounded) self-similar solution when the angle $\beta$ is sufficiently small. We further prove the stability of this self-similar solution. The problem stems from an equation proposed by W. W. Mullins (1957) to model formation of surface grooves on the grain boundaries, where the second boundary condition $v_{xxx}=0$ is replaced by zero slope condition on the curvature of the graph.

For construction of a self-similar solution we solve the initial-boundary problem with homogeneous initial data. However, since the problem is quasilinear and initial data is not compatible with the boundary condition a simple application of an abstract theory for quasilinear parabolic equation is not enough for our purpose. We use a semi-divergence structure to construct a solution.