An analytic semigroup for the magnetoelastic Mindlin–Timoshenko plate model

Abstract

Along with the development of magnetoelastic sensor technology, the subject of magnetoelastic vibrations in plates has attracted considerable interest during the past few decades. More recently, models for the magnetoelastic interactions of a Mindlin–Timoshenko plate and a magnetic field in an electrically conducting plate have been investigated by Grobbelaar-Van Dalsen in [Grobbelaar-Van Dalsen M. On the dissipative effect of a magnetic field in a Mindlin–Timoshenko plate model. Z. Angew. Math. Phys. 2012;63:1047–1065] and [Grobbelaar-Van Dalsen M. Exponential stabilization of magnetoelastic waves in a Mindlin–Timoshenko plate by localized internal damping. Z. Angew. Math. Phys. 2015;66:1751–1776]. In the cited contributions, we accomplished, respectively, polynomial stability of the model, with no mechanical damping, and exponential stability of the model, equipped with nonlinear locally distributed damping that exhibits linear behaviour near the origin. In this article, we complete the cycle by showing that inclusion of Kelvin–Voigt damping in the interior of the plate furnishes the property of analyticity of the semigroup of contractions associated with the model. This provides a direct route to the exponential uniform stability of the model.

Keywords:

• Mindlin–Timoshenko plate
• magnetic field
• analytic semigroup
• Kelvin–Voigt damping
• uniform stability

AMS Subject Classifications:

• 74K10
• 93B15

Notes

No potential conflict of interest was reported by the author.

1 Cf. [52] where Ma et al. established for the undamped model with or without rotational inertia and a nonlinearity of Berger type, polynomial decay of order for the energy associated with the model.