$k$-essence in the DGP brane-world cosmology

We analyze a Dvali-Gabadadze-Porrati (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 behavior of the brane at the low- and the high-energy regimes. The asymptotic behavior 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.