# On curvature flow with driving force under Neumann boundary conditon in the plane

## TL;DR

This paper studies the evolution of axisymmetric curves driven by mean curvature with boundary conditions in the half space, classifying solutions and analyzing fattening phenomena using level set methods.

## Contribution

It introduces criteria for fattening in curvature flow with boundary conditions and classifies solution behaviors in this context.

## Key findings

- Classified solutions into three categories.
- Provided criteria to determine fattening or non-fattening.
- Described asymptotic behaviors for each solution category.

## Abstract

We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the half space. We impose a boundary condition that the curves are perpendicular to the boundary for $t>0$, however, the initial curve intersects the boundary tangentially. In other words, the initial curve is oriented singularly. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.