  In this paper, we discuss singular Neumann boundary problem for a class of
nonlinear parabolic equations in one space dimension. Our boundary problem
describes motion of a planar curve sliding along the boundary with a zero
contact angle, which can be viewed as a limiting model for the capillary
phenomenon. We study the uniqueness and existence of solutions by using the
viscosity solution theory. We also show the convergence of the solution to a
traveling wave as time proceeds to infinity when the initial value is assumed
to be convex.
