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

TL;DR

This paper investigates a singular Neumann boundary problem for nonlinear parabolic equations in one dimension, modeling boundary motion with zero contact angle, and analyzes solution existence, uniqueness, and long-term behavior.

Contribution

It introduces a viscosity solution framework for the problem and proves convergence to traveling waves for convex initial data.

Findings

Existence and uniqueness of solutions established.

Solutions converge to traveling waves over time.

Model describes capillary phenomena with boundary motion.

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. 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.