Stability of the wave equations on a tree with local Kelvin-Voigt damping

TL;DR

This paper investigates the stability of wave equations on a tree-shaped network of elastic strings with localized Kelvin-Voigt damping, establishing conditions for exponential and polynomial decay of energy.

Contribution

It proves stability results for wave equations on a tree with local damping, under specific compatibility and continuity conditions at the vertices.

Findings

Exponential stability under certain conditions

Polynomial stability under weaker conditions

Stability depends on damping coefficient continuity

Abstract

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved.