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Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation. (English) Zbl 1437.35113

This paper deals with the nonlocal rotation-Camassa-Holm equation \[ \begin{cases} m_t+\alpha\varepsilon(um_x +2mu_x)+cu_x ?\beta_0\mu u_{xxx} +w_1\varepsilon^2u^2u_x +w_2\varepsilon^3u^3u_x = 0,&{}\\ m = u- \beta\mu u_{xx},&{} \end{cases} \] that describes the motion of long-crested shallow water waves propagating in one direction with the effect of Earth’s rotation.
The authors carry out an analysis of the characteristic dynamics of \(M = \gamma u - u_x\) and \(N = \gamma u + u_x\). They establish the monotonicity of \(M\) and \(N\), that implies the finite-time wave-breaking. In addition they obtain a persistence result on solutions \(u\) in some weighted \(L^p\) spaces, that implies the spatial asymptotic behavior of a special class of solutions \(u\).

MSC:

35B44 Blow-up in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35Q86 PDEs in connection with geophysics
35Q31 Euler equations
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