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Wave breaking and infinite propagation speed for a modified two-component Camassa-Holm system with \(\kappa = 0\). (English) Zbl 1311.35262

Summary: In this paper, we investigate the modified two-component Camassa-Holm equation with \(\kappa = 0\) on the real line. Firstly, we establish sufficient conditions on the initial data to guarantee that the corresponding solution blows up in finite time for the modified two-component Camassa-Holm (MCH2) system. Then an infinite propagation speed for MCH2 is proved in the following sense: the corresponding solution \(u(x, t) + \kappa\) with compactly supported initial data \((u_0 (x) + \kappa, \rho_0 (x))\) does not have compact \(x\)-support in its lifespan.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35B44 Blow-up in context of PDEs
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