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

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 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.