  We study the well-posedness of the initial value problem for fully nonlinear
evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first
three spatial derivatives of $u.$ We make three primary assumptions about the
form of $f:$ a regularity assumption, a dispersivity assumption, and an
assumption related to the strength of backwards diffusion. Because the third
derivative of $u$ is present in the right-hand side and we effectively assume
that the equation is dispersive, we say that these fully nonlinear evolution
equations are of KdV-type. We prove the well-posedness of the initial value
problem in the Sobolev space $H^{7}(\mathbb{R}).$ The proof relies on gauged
energy estimates which follow after making two regularizations, a parabolic
regularization and mollification of the initial data.
