Existence of non-convex traveling waves for surface diffusion of curves with constant contact angles

TL;DR

This paper investigates the existence, uniqueness, and convexity of traveling wave solutions for surface diffusion of plane curves with fixed contact angles, revealing conditions under which these properties may be lost.

Contribution

It demonstrates that while existence of traveling waves holds for contact angles in (0, π/2), uniqueness and convexity can be lost depending on contact angle conditions.

Findings

Existence of traveling waves for all contact angles in (0, π/2)

Loss of uniqueness depending on contact angle conditions

Loss of convexity depending on contact angle conditions

Abstract

The traveling waves for surface diffusion of plane curves are studied. We consider an evolving plane curve with two endpoints, which can move freely on the x-axis with generating constant contact angles. For the evolution of this plane curve governed by surface diffusion, we discuss the existence, the uniqueness and the convexity of traveling waves. The main results show that the uniqueness and the convexity can be lost in depending on the conditions of the contact angles, although the existence holds for any contact angles in the interval $(0,\pi/2)$.