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Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. (English) Zbl 0934.35153
The Cauchy problem for the periodic Camassa-Holm equation \[ u_t-u_{xxt}+3uu_x=2 u_xu_{xx}+uu_{xxx}, t>0,x\in\mathbb{R},\tag{1} \]
\[ u(x,0)=u_0(x), x\in \mathbb{R},\quad u(x+1,t)= u(x,t),\;t\geq 0,x\in\mathbb{R} \] is studied. First of all it is proved that the problem is locally well-posed for initial data \(u_0\) in \(H^3(S)\), \(S=\mathbb{R}/ \mathbb{Z}\). The Kato’s theory for abstract quasilinear evolution equations of hyperbolic type is used. The solutions can be defined globally or can blow up in finite time, depending on the shape of the initial data. More precisely, a priori estimates are used to prove that if \(y_0\equiv u_0-u_{0,xx}\) does not change sign, then the corresponding solution exists globally. On the contrary, if \(u_0\) is not identically zero and \(\int_Sy_0=0\) or \(\int_S(u^3_0+ u_0u^2_{0,x})=0\) then the corresponding solution blows up in finite time. The last important theorem proved in the paper states that if \(u_0\) is in \(H^1(S)\) and \(y_0\) is a positive Radon measure on \(S\) then there exists a unique global weak solution of (1). Peaked solitary waves are particular cases of such weak solutions.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35L75 Higher-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
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