  We study the initial value problem of fully nonlinear third-order dispersive
equations on the torus. Under some conditions on the nonlinearity and the data,
we prove that the equation behaves like a parabolic one: there exists a unique
local solution in one direction of time that is infinitely smooth and the
problem in not well-posed in the other direction. Under other conditions on the
nonlinearity and the data, we prove that the equation behaves like a dispersive
one: there exists a unique local solution in both directions of time.
