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.