# 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.

## Key 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.