On the supercritical KdV equation with time-oscillating nonlinearity

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 $\varphi \in H^1(\mathbb{R})$ 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(ωt)\partial_x(u^{k+1}) =0, where $g$ is a periodic function and $k\ge 5$ is an integer. We prove that, for given initial data $\varphi \in H^1(\mathbb{R})$, 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 $\Vert U{\Vert }_{L_x^5{L}_t^{10}}<\infty$, then we prove that the solution $u_{\omega }$ is also global provided $|\omega |$ is sufficiently large.

24 Pages