Asymptotic stability and regularity of solutions for a magnetoelastic system in bounded domains

TL;DR

This paper establishes the existence, stability, and asymptotic behavior of solutions for a magnetoelastic system in bounded domains, demonstrating exponential decay of energy and analyzing both 3D and 2D cases with different dissipation mechanisms.

Contribution

It provides new results on the existence and stability of strong time-periodic solutions for magnetoelastic systems, including decay rates and the use of LaSalle invariance principle.

Findings

Existence of strong time-periodic solutions in 3D domains.

Exponential decay of perturbation energy over time.

Asymptotic behavior analysis in 2D domains using LaSalle invariance.

Abstract

We prove the existence of strong time-periodic solutions and their asymptotic stability with the total energy of the perturbations decaying to zero at an exponential decay rate as $t \rightarrow \infty$ for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domain. The mathematical model includes a mechanical dissipation and a periodic forcing function of period $T$. In the second part of the paper, we consider a magnetoelastic system in the form of a semilinear initial boundary value problem in a bounded, simply-connected two-dimensional domain. We use LaSalle invariance principle to obtain results on the asymptotic behavior of solutions. This second result was obtained for the system under the action of only one dissipation (the natural dissipation of the system).