In this talk, we discuss several recent results concerning global stability and large-time behavior of magnetohydrodynamic (MHD) systems with partial or anisotropic dissipation. We first consider incompressible MHD equations in which viscosity and magnetic diffusion act only in selected spatial directions. Although the dissipation is far from the standard isotropic setting, the coupling with a constant background magnetic field generates hidden damped-wave structures that provide additional regularization and stabilization.

We then turn to compressible MHD systems, including non-resistive and partially dissipative models. By exploiting the interaction between acoustic waves, magnetic fields, and fluid motion, we reveal new mechanisms through which magnetic fields recover missing dissipation and prevent nonlinear instability. The analysis combines spectral methods, anisotropic energy estimates, and negative Sobolev techniques.