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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 +σρρ x =0,t>0,x;
ρ t +(ρu) x =0,t>0,x,

which takes the equivalent form of a quasilinear evolution equation of hyperbolic type:

u t +uu x + x G*u 2 +1 2u x 2 +σ 2ρ 2 =0,t>0,x;
ρ t +(ρu) x =0,t>0,x,

where the sign * denotes the spatial convolution, G(s) is the associated Green function of the operator (1- x 2 ) -1 , σ 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.

35G25Initial value problems for nonlinear higher-order PDE
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B44Blow-up (PDE)
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