# Mean curvature flow with driving force on fixed extreme points

## TL;DR

This paper studies the evolution of curves under mean curvature flow with an external driving force, focusing on fixed endpoints, providing existence, uniqueness, classification, and asymptotic behavior of solutions.

## Contribution

It introduces a local existence and uniqueness result for the flow with fixed endpoints and classifies solutions for specific initial curves, detailing their long-term behavior.

## Key findings

- Established local existence and uniqueness for $C^2$ initial curves.
- Classified solutions into three categories based on initial conditions.
- Described asymptotic behaviors for each solution category.

## Abstract

In this paper, we consider the mean curvature flow with driving force on fixed extreme points in the plane. We give a general local existence and uniqueness result of this problem with $C^2$ initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.