  We study a flat 3-brane in presence of a linear $k$ field with nonzero
cosmological constant $\Lambda_{4}$. In this model the crossing of the phantom
divide (PD) occurs when the $k$-essence energy density becomes negative. We
show that in the high energy regime the effective equation of state has a
resemblance of a modified Chaplygin gas while in the low energy regime it
becomes linear. We find a scale factor that begins from a singularity and
evolves to a de Sitter stable stage while other solutions have a
super-accelerated regime and end with a big rip. We use the energy conditions
to show when the effective equation of state of the brane-universe crosses the
PD.
