  This work is concerned about the Cauchy problem for the following generalized
KdV- Burgers equation
  \begin{equation*}
  \left\{\begin{array}{l}
  \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0,
  u(0,\,x)=u_0(x).
  \end{array}
  \right.
  \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using
Besov-Bourgain Spaces, we establish a bilinear estimate and following the
framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global
well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness
in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.
