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Wave-breaking phenomena for a weakly dissipative shallow water equation. (English) Zbl 1440.35284

Summary: Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms \(u^{2n + 1}\) and \(u^{2m} u_{xx}\), the equation loses the conservation law \(E = \int\nolimits_{\mathbb{S}} (u^2 + u^2_x) \text{d}x.\) This difficulty has been dealt with by establishing the energy inequality.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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