  We consider multiple k-essence sources and obtain the conditions their
kinetic functions must satisfy so that purely kinetic k-essences lead to models
with phantom barrier crossing. After that, we show that polynomial kinetic
functions allow the integration of the dynamical equations determining the
geometry and the k-fields. The models thus obtained have accelerated expansion
at late cosmic time, representing universes ending at a finite time with a big
rip singularity. In addition, these models begin to evolve from an initial
singularity, so they have a finite time span.

Astrophysics