k-essence in the DGP brane-world cosmology

We analyse a DGP brane filled with a k-essence field and assume the k-field evolving linearly with the cosmic time of the brane. We then solve analytically the Friedmann equation and deduce the different behaviour of the brane at the low and the high energy regimes. The asymptotic behaviour can be quite different involving accelerating branes, big bangs, big crunches, big rips or quiescent singularities. The latter correspond to a type of sudden singularity.