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Blow-up and global solutions to a new integrable model with two components. (English) Zbl 1205.35045

The author considers the two-component generalized Camassa-Holm system
\[ u_{t}-u_{xxt}+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}+\sigma \rho\rho_{x}=0,\quad t>0,\;x\in\mathbb R; \]
\[ \rho_{t}+(\rho u)_{x}=0,\quad t>0,\;x\in\mathbb R, \]
which takes the equivalent form of a quasilinear evolution equation of hyperbolic type:
\[ u_{t}+uu_{x}+\partial_{x} \left(G\ast \left(u^{2}+\tfrac{1}{2}u_{x}^{2}+\tfrac{\sigma}{2}\rho^{2}\right)\right)=0, \quad t>0,\;x\in\mathbb R; \]
\[ \rho_{t}+(\rho u)_{x}=0,\quad t>0,\;x\in\mathbb R, \]
where the sign \(\ast\) denotes the spatial convolution, \(G(s)\) is the associated Green function of the operator \((1-\partial_{x}^{2})^{-1},\) \(\sigma\) can be chosen to 1 or \(-1\). For this system, the global existence and blow-up phenomena questions are investigated. The blow-up criteria for the nonperiodic case are also obtained.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B44 Blow-up in context of PDEs
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