  Considered is the generalized Korteweg-de Vries-Burgers equation $$
u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\alpha}u=0,\quad t\in \mathbb{R}^{+}, x\in
\mathbb{R}, $$ with $0\leq \alpha\le 1$. We prove a sharp results on the
associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For
$s>-\min\{\frac {3+2\alpha}4, 1\}$ we give the well-posedness of solutions of
the Cauchy problem, while for $\frac 12\le\alpha\le 1$ and for $s<-\min\{\frac
{3+2\alpha}4, 1\}$ we show some ill-posedness issues.
