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KdV, BO and friends in weighted Sobolev spaces. (English) Zbl 0717.35076
Functional-analytic methods for partial differential equations, Proc. Conf. Symp., Tokyo/Jap. 1989, Lect. Notes Math. 1450, 104-121 (1990).
[For the entire collection see Zbl 0707.00017.]
The purpose of this paper is to discuss some aspects of the relationship between differentiability and special decay of the real-valued solutions of the Cauchy problem for certain nonlinear evolution equations. We will concentrate on the equations of Korteweg-de Vries (KDV) and Benjamin-Ono (BO) but we will also consider the less known, albeit very interesting, equation of Smith (S). Thus, we will study some properties of the following problems: \[ \begin{alignedat}{3} \partial_ tu &= -\partial_ x(u^ 2+\partial^ 2_ xu), &\;u(0) &= \phi,\tag{KDV} \\ \partial_ tu &= - \partial_ x(u^ 2+2\sigma \partial_ xu), &\;u(0) &= \phi,\tag{BO} \\ \partial_ tu &= -\partial_ x(u^ 2+2Lu), &\;u(0) &= \phi,\tag{S}\end{alignedat} \] where \(\sigma\) denotes the Hilbert transform \[ (\sigma f)(x)=p\cdot v\cdot (1/\pi)\int_{{\mathbb{R}}}(f(y)/(y-x))dy \] and L is the operator given by \(Lf=(\sqrt{-\partial^ 2_ x+1}-1)f\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)