Advances in Differential Equations

The Cauchy problem for a generalized Camassa-Holm equation

A. Alexandrou Himonas and Curtis Holliman

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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.

Article information

Adv. Differential Equations, Volume 19, Number 1/2 (2014), 161-200.

First available in Project Euclid: 12 November 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]


Himonas, A. Alexandrou; Holliman, Curtis. The Cauchy problem for a generalized Camassa-Holm equation. Adv. Differential Equations 19 (2014), no. 1/2, 161--200.

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