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2004 Local and global well-posedness for a class of nonlinear dispersive equations
Radu C. Cascaval
Adv. Differential Equations 9(1-2): 85-132 (2004). DOI: 10.57262/ade/1355867969

Abstract

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form \[ u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R}, \] where $u=u(x,t)$, $x\in \mathbb{R} $ or $x\in \mathbb{T} $. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $ \beta \ge \frac12 $ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1 $), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty $, for some $p < 4\beta $. Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein--Oharu--Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e., $x\in\mathbb{T} $), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

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Radu C. Cascaval. "Local and global well-posedness for a class of nonlinear dispersive equations." Adv. Differential Equations 9 (1-2) 85 - 132, 2004. https://doi.org/10.57262/ade/1355867969

Information

Published: 2004
First available in Project Euclid: 18 December 2012

zbMATH: 1103.35089
MathSciNet: MR2099607
Digital Object Identifier: 10.57262/ade/1355867969

Subjects:
Primary: 35Q53
Secondary: 34G20 , 35B30 , 47H20 , 47N20

Rights: Copyright © 2004 Khayyam Publishing, Inc.

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