  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.
