On the supercritical KdV equation with time-oscillating nonlinearity

TL;DR

This paper studies the supercritical gKdV equation with a time-oscillating nonlinearity, proving that solutions converge to an averaged equation as oscillation frequency increases and establishing conditions for global existence.

Contribution

It introduces an analysis of the supercritical gKdV with periodic time-dependent nonlinearity, showing convergence to an averaged model and conditions for global solutions at high oscillation frequencies.

Findings

Solutions converge to the averaged equation as frequency increases.

High-frequency oscillations do not prevent global existence under certain conditions.

The averaged nonlinearity is the mean of the periodic function g.

Abstract

For the initial value problem (IVP) associated the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, numerical evidence \cite{BDKM1, BSS1} shows that there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation \cite{ACKM, KP}, the physicists claim that a periodic time dependent term in factor of the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\R)$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, 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.