K-dynamics: well-posed 1+1 evolutions in K-essence

Abstract 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. restrict to spherical symmetry show there exists a large class of no breakdown occurs outside apparent black hole horizons, even in presence scalar shocks/caustics, except small set initial data sufficiently close critical collapse. characterise these problematic data, they lead or diverging (coordinate) speeds characteristic curves. discuss physical relevance this propose ways overcome it.