  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).
