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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Global existence for the critical generalized KdV equation


Authors: G. Fonseca, F. Linares and G. Ponce
Journal: Proc. Amer. Math. Soc. 131 (2003), 1847-1855
MSC (2000): Primary 35Q53
Published electronically: November 6, 2002
MathSciNet review: 1955273
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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,

\begin{displaymath}u_t+u_{xxx}+u^4\,u_x=0,\quad x,\,t\in\mathbb{R} .\end{displaymath}

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 $\Vert u_0\Vert _{L^2}<\Vert Q\Vert _{L^2}$, where $Q$ is the solitary wave solution of the equation.


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Additional Information

G. Fonseca
Affiliation: Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Email: gfonseca@matematicas.unal.edu.co

F. Linares
Affiliation: Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brazil
Email: linares@impa.br

G. Ponce
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: ponce@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06871-5
PII: S 0002-9939(02)06871-5
Received by editor(s): January 30, 2002
Published electronically: November 6, 2002
Additional Notes: The first author was partially supported by DIB-Universidad Nacional de Colombia
The second author was partially supported by CNP-q Brazil
The third author was partially supported by an NSF grant
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society