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2006 Large data wave operator for the generalized Korteweg-de Vries equations
Raphaël Côte
Differential Integral Equations 19(2): 163-188 (2006).

Abstract

We consider the generalized Korteweg-de Vries equations : \[ u_t + (u_{xx} + u^p)_x =0, \quad t,x \in \mathbb R, \] for $p \in (3,\infty)$. Let $U(t)$ be the associated linear group. Given $V$ in the weighted Sobolev space $H^{2,2} = \{ f \in L^2 : (1+|x|)^2(1-\partial_x^2)f \|_{L^2} < \infty\}$, possibly large, we construct a solution $u(t)$ of the generalized Korteweg-de Vries equation such that : \[ \lim_{t \to \infty} \| u(t) - U(t) V \|_{H^1} =0. \] We also prove uniqueness of such a solution in an adequate space. In the $L^2$-critical case ($p=5$), this result can be improved to any possibly large function $V$ in $L^2$ (with convergence in $L^2$).

Citation

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Raphaël Côte. "Large data wave operator for the generalized Korteweg-de Vries equations." Differential Integral Equations 19 (2) 163 - 188, 2006.

Information

Published: 2006
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35408
MathSciNet: MR2194502

Subjects:
Primary: 35Q53
Secondary: 35B40

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.19 • No. 2 • 2006
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