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

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

2007-06-21

arxiv.org/abs/0706.3091v1

Mathematics

Analysis of PDEs

20 pages

Scientific paper

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.

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