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Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. (English) Zbl 0808.35128
The well-posedness, which includes existence, uniqueness, and continuous dependence upon initial data, is studied for the initial-value problem for the generalized Korteweg-de Vries (KdV) equations of the form $$u\sb t+ u\sb{xxx}+ u\sp k u\sb x=0, \tag 1$$ where $k$ is a positive integer. It is well known that equation (1) is exactly integrable for $k=1$ (the classical KdV equation) and for $k=2$ (the modified KdV equation). At any value of $k$, equation (1) has two fundamental integrals of motion, the “momentum” $\int\sb{-\infty}\sp{+\infty} u\sp 2(x)dx,$ and the energy (Hamiltonian). These two integrals of motion are essentially used in the well-posedness analysis. The analysis is based on global estimates for an explicit solution of the linear initial-value problem associate to equation (1), $v\sb t+ v\sb{xxx}=0$ combined with the contraction mapping principle. Introducing special functional norms, the authors demonstrate that, for each particular value of $k$, there is a relevant class of the Sobolev space to which the initial data $u\sb 0(x)\equiv u(x,t=0)$ must belong in order to allow for the well-posedness proof, the existence being proved for finite times. The technique of the proof and the particular results obtained are essentially different for the cases $k<4$, $k=4$, and $k>4$, which reflects the fundamental property of equation (1): while at $k<4$ the solution developing from generic initial data remains smooth indefinitely long, the weak and strong collapse sets in at a finite time, respectively, at $k=4$ and at $k>4$.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35P25 Scattering theory (PDE) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B45 A priori estimates for solutions of PDE
##### Keywords:
well-posedness of KdV equations; collapse; Sobolev spaces
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