*(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

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” ${\int}_{-\infty}^{+\infty}{u}^{2}\left(x\right)dx,$ 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), ${v}_{t}+{v}_{xxx}=0$ 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 ${u}_{0}\left(x\right)\equiv u(x,t=0)$ 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 $k>4$.