  We investigate the issue of uniqueness of the limit flow for a relevant class
of quasi-linear parabolic equations defined on the whole space. More precisely,
we shall investigate conditions which guarantee that the global solutions decay
at infinity uniformly in time and their entire trajectory approaches a single
steady state as time goes to infinity. Finally, we obtain a characterization of
solutions which blow-up, vanish or converge to a stationary state for initial
data of the form $\lambda \varphi_0$ while $\lambda>0$ crosses a bifurcation
value $\lambda_0$.
