  We investigate local well-posedness of the initial value problem for Lovelock
and Horndeski theories of gravity. A necessary condition for local
well-posedness is strong hyperbolicity of the equations of motion. Even weak
hyperbolicity can fail for strong fields so we restrict to weak fields. The
Einstein equation is known to be strongly hyperbolic in harmonic gauge so we
study Lovelock theories in harmonic gauge. We show that the equation of motion
is always weakly hyperbolic for weak fields but, in a generic weak-field
background, it is not strongly hyperbolic. For Horndeski theories, we prove
that, for weak fields, the equation of motion is always weakly hyperbolic in
any generalized harmonic gauge. For some Horndeski theories there exists a
generalized harmonic gauge for which the equation of motion is strongly
hyperbolic in a weak-field background. This includes "k-essence" like theories.
However, for more general Horndeski theories, there is no generalized harmonic
gauge for which the equation of motion is strongly hyperbolic in a generic
weak-field background. Our results show that the standard method used to
establish local well-posedness of the Einstein equation does not extend to
Lovelock or general Horndeski theories. This raises the possibility that these
theories may not admit a well-posed initial value problem even for weak fields.
