Uniqueness of limit flow for a class of quasi-linear parabolic equations

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 $\lambda_0$.