  We study the vacuum Cauchy problem for K-essence, i.e. cosmologically
relevant scalar-tensor theories that involve first-order derivative
self-interactions, and which pass all existing gravitational wave bounds. We
restrict to spherical symmetry and show that there exists a large class of
theories for which no breakdown of the Cauchy problem occurs outside apparent
black hole horizons, even in the presence of scalar shocks/caustics, except for
a small set of initial data sufficiently close to critical black hole collapse.
We characterise these problematic initial data, and show that they lead to
large or even diverging (coordinate) speeds for the characteristic curves. We
discuss the physical relevance of this problem and propose ways to overcome it.
