Sharp ill-posedness and well-posedness results for dissipative KdV equations on the real line

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 $H^{-p\mathrm{/}2}(I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}R)$ and sharp ill-posedness in $H^s(I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}R)$ when $s<-p\mathrm{/}2$ with $p\ge 2$.