Can dark energy evolve to the phantom?

Dark energy with the equation of state $w(z)$ rapidly evolving from the dustlike ($w\simeq 0$ at $z\sim 1$) to the phantomlike ($-1.2\lesssim w\lesssim -1$ at $z\simeq 0$) has been recently proposed as the best fit for the supernovae Ia data. Assuming that a dark energy component with an arbitrary scalar-field Lagrangian $p(\phi ,{\nabla }_{\mu }\phi )$ dominates in the flat Friedmann universe, we analyze the possibility of a dynamical transition from the states $(\phi ,\dot{\phi })$ with $w\ge -1$ to those with $w<-1$ or vice versa. We have found that generally such transitions are physically implausible because they are either realized by a discrete set of trajectories in the phase space or are unstable with respect to the cosmological perturbations. This conclusion is confirmed by a comparison of the analytic results with numerical solutions obtained for simple models. Without the assumption of the dark energy domination, this result still holds for a certain class of dark energy Lagrangians, in particular, for Lagrangians quadratic in ${\nabla }_{\mu }\phi$. The result is insensitive to topology of the Friedmann universe as well.