Well-posedness of the initial boundary value problem for the motion of an inextensible hanging string

TL;DR

This paper proves local well-posedness for the nonlinear, nonlocal hyperbolic equations governing the motion of an inextensible hanging string under gravity, addressing degeneracy at the free end.

Contribution

It establishes well-posedness of the initial boundary value problem for the string's motion in weighted Sobolev spaces at the quasilinear regularity threshold, under a stability condition.

Findings

Well-posedness shown in weighted Sobolev spaces

Degeneracy at the free end addressed

Results extend previous work on string dynamics

Abstract

We consider the motion of an inextensible hanging string of finite length under the action of the gravity. The motion is governed by nonlinear and nonlocal hyperbolic equations, which is degenerate at the free end of the string. We show that the initial boundary value problem to the equations of motion is well-posed locally in time in weighted Sobolev spaces at the quasilinear regularity threshold under a stability condition. This paper is a continuation of our preceding articles.