Global existence and blow-up solutions for a nonlinear shallow water equation
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Considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by Dullin, Gottward and Holm. Blow-up can occur only in the form of wave-breaking. A wave-breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow-up rate is established. The blow-up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined.
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