# Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping

## TL;DR

This paper investigates the long-term decay behavior of coupled Euler-Bernoulli plate and wave equations with localized Kelvin-Voigt damping, showing solutions decay logarithmically at infinity despite partial stabilization.

## Contribution

It introduces a novel frequency method combined with Carleman estimates to analyze the asymptotic decay of coupled PDEs with localized damping.

## Key findings

- Solutions decay logarithmically at infinity
- Partial stabilization influences the decay rate
- The method effectively analyzes resolvent behavior

## Abstract

Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.