On the supercritical KdV equation with time-oscillating nonlinearity

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For the initial value problem (IVP) associated the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, uₜ+∂ₓ³u+∂ₓ(uᵏ⁺¹) =0, k≥ 5, numerical evidence BDKM1, BSS1 shows that there are initial data ϕ∈ H¹(R) such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation 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 uₜ+∂ₓ³u+g(ωt)∂ₓ(uᵏ⁺¹) =0, where g is a periodic function and k≥ 5 is an integer. We prove that, for given initial data ϕ∈ H¹(), as |ω|→ ∞, the solution uω converges to the solution U of the initial value problem associated to Uₜ+∂ₓ³U+m(g)∂ₓ(Uᵏ⁺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 Uₗₓ₅ₗₜ¹⁰<∞, then we prove that the solution uω is also global provided |ω| is sufficiently large.