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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 t +u xxx +u k u x =0,(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” - + u 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 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 (x)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:
35Q53KdV-like (Korteweg-de Vries) equations
35P25Scattering theory (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B45A priori estimates for solutions of PDE