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May/June 2011 On the critical KdV equation with time-oscillating nonlinearity
X. Carvajal, M Panthee, M. Scialom
Differential Integral Equations 24(5/6): 541-567 (May/June 2011).

Abstract

We investigate the initial-value problem (IVP) associated with the equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0, \end{equation*} where $g$ is a periodic function. We prove that, for given initial data $\phi \in H^1(\mathbb R)$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial-value problem associated with \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^5) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large.

Citation

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X. Carvajal. M Panthee. M. Scialom. "On the critical KdV equation with time-oscillating nonlinearity." Differential Integral Equations 24 (5/6) 541 - 567, May/June 2011.

Information

Published: May/June 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35261
MathSciNet: MR2809621

Subjects:
Primary: 35Q35, 35Q53

Rights: Copyright © 2011 Khayyam Publishing, Inc.

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Vol.24 • No. 5/6 • May/June 2011
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