Abstract
For the initial value problem (IVP) associated to the generalized Korteweg–de Vries (gKdV) equation with supercritical nonlinearity,
numerical evidence [3] 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 [1, 18], it has been claimed that a periodic time dependent coefficient in 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
where g is a periodic function and \({k\geq 5}\) is an integer. We prove that, for given initial data \({\phi \in H^1(\mathbb{R})}\), as \({|\omega|\to \infty}\), the solution \({u_{\omega} }\) converges to the solution U of the initial value problem associated to
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 \({\|U\|_{L_x^{5}L_t^{10}}<\infty}\), then we prove that the solution \({u_{\omega} }\) is also global provided \({|\omega|}\) is sufficiently large.
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M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil.
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Panthee, M., Scialom, M. On the supercritical KdV equation with time-oscillating nonlinearity. Nonlinear Differ. Equ. Appl. 20, 1191–1212 (2013). https://doi.org/10.1007/s00030-012-0204-z
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DOI: https://doi.org/10.1007/s00030-012-0204-z