Stability analysis of degenerately-damped oscillations

TL;DR

This paper investigates the well-posedness and stability of a PDE modeling a vibrating string with degenerately-damped behavior, revealing energy decay properties and providing numerical and analytical insights into the system's dynamics.

Contribution

It introduces a novel analysis of a degenerately damped PDE, showing energy decay and stability characteristics, supported by numerical simulations and explicit solutions.

Findings

Energy decreases monotonically over time

Lower-order norms decay uniformly

Solutions can be explicitly characterized for certain initial data

Abstract

Presented here is a study of well-posedness and asymptotic stability of a "degenerately damped" PDE modeling a vibrating elastic string. The coefficient of the damping may vanish at small amplitudes thus weakening the effect of the dissipation. It is shown that the resulting dynamical system has strictly monotonically decreasing energy and uniformly decaying lower-order norms, however, is not uniformly stable on the associated finite-energy space. These theoretical findings were motivated by numerical simulations of this model using a finite element scheme and successive approximations. A description of the numerical approach and sample plots of energy decay are supplied. In addition, for certain initial data the solution can be determined in closed form up to a dissipative nonlinear ordinary differential equation. Such solutions can be used to assess the accuracy of the numerical examples.