Well-posedness of the initial boundary value problem for degenerate hyperbolic systems with a localized term and its application to the linearized system for the motion of an inextensible hanging string

TL;DR

This work investigates the well-posedness of a degenerate hyperbolic initial boundary value problem in one spatial dimension with a localized coupling term, motivated by the linearized motion of an inextensible hanging string. The authors develop a weighted Sobolev framework, derive robust energy estimates, and employ a regularization plus a unit-disk transformation to establish existence and uniqueness (Theorem Th1) for the degenerate problem. They then address the linearized system arising from the hanging-string model by decomposing the tension variation into a principal and a lower-order part, proving well-posedness (Theorem Th2) with detailed control of the localized term and higher-order regularity via additional assumptions. The results provide a solid foundation for analyzing the full nonlinear string dynamics and demonstrate how localized couplings can be handled within a degenerate hyperbolic IBVP setting. Collectively, the work advances the mathematical understanding of degenerate hyperbolic systems with localized terms and their applications to elastic/string models at finite length.

Abstract

Motivated by an analysis on the well-posedness of the initial boundary value problem for the motion of an inextensible hanging string, we first consider an initial boundary value problem for one-dimensional degenerate hyperbolic systems with a localized term and show its well-posedness in weighted Sobolev spaces. We then consider the linearized system for the motion of an inextensible hanging string. Well-posedness of its initial boundary value problem is demonstrated as an application of the result obtained in the first part.