Can dark energy evolve to the phantom?

Dark energy with the equation of state $w(z)$ rapidly evolving from the dustlike ($w\ensuremath{\simeq}0$ at $z\ensuremath{\sim}1$) to the phantomlike ($\ensuremath{-}1.2\ensuremath{\lesssim}w\ensuremath{\lesssim}\ensuremath{-}1$ at $z\ensuremath{\simeq}0$) 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 $p(\ensuremath{\varphi},{\ensuremath{\nabla}}_{\ensuremath{\mu}}\ensuremath{\varphi})$ dominates in the flat Friedmann universe, we analyze the possibility of a dynamical transition from the states $(\ensuremath{\varphi},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\varphi}})$ with…