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January/February 2014 The Cauchy problem for a generalized Camassa-Holm equation
A. Alexandrou Himonas, Curtis Holliman
Adv. Differential Equations 19(1/2): 161-200 (January/February 2014).

Abstract

This work studies the initial value problem (i.v.p.) for a generalized Camassa-Holm equation with $(k+1)$-order nonlinearities (g-$k$bCH) and containing, as its members, three integrable equations: the Camassa-Holm, the Degasperis-Procesi and the Novikov equations. For $s>3/2$, using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$ on both the circle and the line in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates.

Citation

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A. Alexandrou Himonas. Curtis Holliman. "The Cauchy problem for a generalized Camassa-Holm equation." Adv. Differential Equations 19 (1/2) 161 - 200, January/February 2014.

Information

Published: January/February 2014
First available in Project Euclid: 12 November 2013

zbMATH: 1285.35093
MathSciNet: MR3161659

Subjects:
Primary: 35Q53

Rights: Copyright © 2014 Khayyam Publishing, Inc.

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Vol.19 • No. 1/2 • January/February 2014
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