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On the Cauchy problem for the periodic Camassa-Holm equation. (English) Zbl 0889.35022
We consider the Cauchy problem for the recently derived shallow water equation \[ u_t- u_{txx} +3uu_x =2u_xu_{xx} +uu_{xxx},\;t>0,\;x\in\mathbb{R}, \quad u(0,x) =u_0(x), \] where \(u\) describes the free surface of the water above a flat bottom. With \(m=u- u_{xx}\), the equation can be written as \[ m_t= -2mu_x- m_xu,\;t>0,\;x\in\mathbb{R}, \quad m (0)= \varphi. \tag{1} \] We look for solutions of (1) which are spatially of period 1. We will prove a local existence theorem for (1) in the Sobolev space \(H^2\) of functions with period 1, using Kato’s method for abstract quasilinear equations. If the initial data \(\varphi\in H^2_p\) do not change sign \((\varphi\geq 0\) or \(\varphi\leq 0)\), we prove that (1) is globally well-posed in \(H^2_p\). We also show that for a large class of initial data \(\varphi\in H^2_p\) taking both strictly positive and strictly negative values, the solution of (1) blows up in finite time.

MSC:
35G25 Initial value problems for nonlinear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35F25 Initial value problems for nonlinear first-order PDEs
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