Global existence and blow-up for a shallow water equation.

*(English)*Zbl 0918.35005An 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:

$$\left\{\begin{array}{cc}{u}_{t}-{u}_{xxt}+3u{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx},\phantom{\rule{2.em}{0ex}}t>0,\phantom{\rule{1.em}{0ex}}\hfill & x\in \mathbb{R},\hfill \\ u(0,x)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}\hfill & x\in \mathbb{R}\xb7\hfill \end{array}\right.\phantom{\rule{2.em}{0ex}}\left(1\right)$$

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 waterâ€™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. In addition, we introduce the notion of weak solutions to (1) suitable for soliton interaction.

##### MSC:

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35Q35 | PDEs in connection with fluid mechanics |

35B40 | Asymptotic behavior of solutions of PDE |

35Q53 | KdV-like (Korteweg-de Vries) equations |