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On the critical KdV equation with time-oscillating nonlinearity

We investigate the initial value problem (IVP) associated to the equation ut + ∂ xu+ g(ωt)∂x(u ) = 0, where g is a periodic function. We prove that, for given initial data φ ∈ H(R), as |ω| → ∞, the solution uω converges to the solution U of the initial value problem associated to Ut + ∂ xU +m(g)∂x(U ) = 0, with the same initial data, wherem(g) is the average of the periodic function g. Moreover, if the solution U is global and satisfies ‖U‖L5xL10 t < ∞, then we prove that the solution uω is also global provided |ω| is sufficiently large.