Mathematical Research Letters

Volume 18 (2011)

Number 2

The supercritical generalized KdV equation: Global well-posedness in the energy space and below

Pages: 357 – 377



Luiz G. Farah (ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Caixa Postal 702, 30123-970, Belo Horizonte-MG, Brazil)

Felipe Linares (IMPA, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil)

Ademir Pastor (IMECC-UNICAMP, Rua Sérgio Buarque de Hol)


We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$ belongs to $H^1(\R)$ and satisfies $E(u_0)^{s_k} M(u_0)^{1-s_k} <E(Q)^{s_k} M(Q)^{1-s_k}$, $E(u_0)\geq0$, and $\|\partial_x u_0\|_{L^2}^{s_k}\|u_0\|_{L^2}^{1-s_k} <\|\partial_x Q\|_{L^2}^{s_k}\|Q\|_{L^2}^{1-s_k}$, where $M(u)$ and $E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\R)$. Here, $s_k=\frac{(k-4)}{2k}$ and $Q$ is the ground state solution corresponding to the gKdV equation. In the defocusing case ($\mu=-1$), if $k$ is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces $H^s(\mathbb{R})$, $s >\frac{4(k-1)}{5k}$.

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