Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension

Abstract

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. Under a convexity assumption on the initial value, we also show the convergence of the solution to a traveling wave as time proceeds to infinity.