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Global solutions and blow-up phenomena to a shallow water equation. (English) Zbl 1198.35041
A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. By applying the Kato theorem the local well-posedness of solutions for the nonlinear equation in the Sobolev space H s () with s>3/2 is developed. Provided that (1- χ 2 )u 0 does not change sign, u 0 H s (s>3/2) and u 0 L 1 (), the existence and uniqueness of the global solutions to the equation are shown to be true in C([0,);H s ())C 1 ([0,);H s-1 ()). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.
MSC:
35B44Blow-up (PDE)
35G25Initial value problems for nonlinear higher-order PDE
35L05Wave equation (hyperbolic PDE)
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