## Differential and Integral Equations

### On the critical KdV equation with time-oscillating nonlinearity

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 24, Number 5/6 (2011), 541-567.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356018918

Mathematical Reviews number (MathSciNet)
MR2809621

Zentralblatt MATH identifier
1249.35261

#### Citation

Carvajal, X.; Panthee, M; Scialom, M. On the critical KdV equation with time-oscillating nonlinearity. Differential Integral Equations 24 (2011), no. 5/6, 541--567. https://projecteuclid.org/euclid.die/1356018918