Constantin, Adrian; Escher, Joachim Global existence and blow-up for a shallow water equation. (English) Zbl 0918.35005 Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No. 2, 303-328 (1998). An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. Recently, R. Camassa and D. Holm proposed a new model for the same phenomenon: \[ \begin{cases} u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx},\qquad t>0,\quad & x\in\mathbb{R},\\ u(0, x)= u_0(x),\quad & x\in\mathbb{R}.\end{cases}\tag{1} \] The variable \(u(t,x)\) in (1) represents the fluid velocity at time \(t\) in the \(x\) direction in appropriate nondimensional units (or, equivalently, the height of the water’s free surface above a flat bottom).The aim of this paper is to prove local well-posedness of strong solutions to (1) for a large class of initial data, and to analyze global existence and blow-up phenomena. 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