This chapter presents some typical aspects of the qualitative analysis of parabolic equations. It shows how the boundedness of solutions can be obtained together with its importance for the existence of global solutions. The chapter performs a stability analysis of a parabolic P.D.E. arising in anisotropic magneto-hydrodynamics, representing an inhomogeneous diffusion. It devotes to the energy stability of viscous flows in bounded and unbounded regions respectively. The modern version of the method of nonlinear stability, in the L2-norm (“energy method”), of isothermal, viscous incompressible fluid motions in regions which are bounded at least in one direction, was introduced originally by Serrin in 1959. The energy method is well established within the theory of the Liapunov direct method, and can give results which are useful in practice.