# Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping

## TL;DR

This paper investigates the energy decay rates of coupled elastic beam and wave systems with local Kelvin-Voigt damping, establishing polynomial decay estimates and demonstrating the absence of exponential decay.

## Contribution

It provides new polynomial decay estimates for coupled systems with local damping and shows that exponential decay does not occur in these models.

## Key findings

- Energy decays polynomially over time for both damping cases.
- Damping acting on either the beam or wave equation leads to polynomial stability.
- Exponential decay is not achieved in these coupled systems.

## Abstract

We consider a beam and a wave equations coupled on an elastic beam through transmission conditions. The damping which is locally distributed acts through one of the two equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the beam equation. Using a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups we provide a precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.