A New Insight Into the Exponential Stabilization of Viscoelastic Timoshenko Systems With Fourier’s Law

Abstract

This paper presents the surprising result of the exponential stabilization of a viscoelastic Timoshenko beam system with Fourier’s law, regardless of the wave speed equality condition. First, we prove the well-posedness of the system using the Lumer–Phillips theorem. Then, in the direction of the Gearhart–Huang–Prüss theorem, we prove the inclusion $i\mathbb{R}\subset \varrho (\mathcal{A})$ and the boundedness of $(i\lambda I-\mathcal{A}{)}^{-1}$ . This result contrasts with the nature of the Timoshenko system, as inferred from the works of Malacarne and Muñoz Rivera (2016) and Cui and Chai (2023), since the expected result would be the lack of exponential stability. However, we prove exponential decay independent of the wave speed equality relation.