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Higher-order nonlinear dispersive equations

Authors: Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal: Proc. Amer. Math. Soc. 122 (1994), 157-166
MSC: Primary 35G25; Secondary 35Q53
MathSciNet review: 1195480
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Abstract: We study nonlinear dispersive equations of the form

$\displaystyle {\partial _t}u + \partial _x^{2j + 1}u + P(u,{\partial _x}u, \ldots ,\partial _x^{2j}u) = 0,\qquad x,t \in \mathbb{R},\quad j \in {\mathbb{Z}^ + },$

where $ P( \cdot )$ is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size.

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Keywords: Higher order models, smoothing effects, gauge transformation
Article copyright: © Copyright 1994 American Mathematical Society

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