On the well-posedness of the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation

Considered is the generalized Korteweg-de Vries-Burgers equation $$u_t+u_{xxx}+u{u}_x+\mathrm{\mid }{D}_x{\mathrm{\mid }}^{2\alpha }u=0,t\in {\mathbb{R}}^{+},x\in \mathbb{R},$$ with $0\le \alpha \le 1$. 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 $\frac{1}{2}\le \alpha \le 1$ and for $s<-\min{\frac {3+2\alpha}4, 1}$ we show some ill-posedness issues.