Numerical study of a Transmission Problem in Elasticity with kind damping

TL;DR

This paper analyzes a transmission problem in elasticity with damping, showing polynomial decay of solutions over time and validating the decay rates through numerical simulations.

Contribution

It provides a theoretical analysis of decay rates in a damping transmission problem and confirms the results with numerical validation.

Findings

Semigroup exhibits polynomial decay towards zero

Decay rate is proven to be optimal

Numerical study confirms theoretical decay rates

Abstract

We investigate a transmission problem featuring a specific type of damping. Our primary focus is on analyzing the asymptotic behavior of the associated semigroup, $({\mathcal S}_{\mathcal A}(t))_{t\geq 0}$. We demonstrate that this semigroup exhibits a polynomial rate of decay towards zero when the initial data is taken over the domain ${\mathcal D}({\mathcal A})$. Furthermore, we establish that this decay rate is optimal. To support our theoretical findings, we present a comprehensive numerical study that validates and illustrates the sharpness of the obtained decay rates.