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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
##### Keywords:
blow up; shallow water equation
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##### References:
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