Global existence for the critical generalized KdV equation

Fonseca, G.; Linares, F.; Ponce, G.

doi:10.1090/S0002-9939-02-06871-5

Global existence for the critical generalized KdV equation
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by G. Fonseca, F. Linares and G. Ponce PDF

Proc. Amer. Math. Soc. 131 (2003), 1847-1855 Request permission

Abstract:

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation, \[ u_t+u_{xxx}+u^4 u_x=0,\quad x, t\in \mathbb {R}.\] The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space $H^1(\mathbb {R})$, i.e. solutions corresponding to data $u_0\in H^s(\mathbb {R})$, $s>3/4$, with $\|u_0\|_{L^2}<\|Q\|_{L^2}$, where $Q$ is the solitary wave solution of the equation.

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