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On the blow-up scenario for the generalized Camassa-Holm equation. (English) Zbl 1068.35030
The Cauchy problem for the generalized Camassa-Holm (gCH) equation $$ \cases u_t - u_{xx} + ku_x + [g(u)/ 2]_x = 2u_x u_{x} + u_{xxx} ,&t > 0,\;x \in {\Bbb R}, \\ u\left( {0,x} \right) = u_0 (x),&x \in {\Bbb R}, \endcases$$ where $g:{\Bbb R} \to {\Bbb R}$ is given $C^3$-function and $k \in {\Bbb R}$ is a fixed constant, is considered in the paper. The aim of this paper is to address the question of the formation of singularities in finite time for solutions of gCH. By applying a recent result for the time evolution of the extrema of a function and looking at the time-evolution of the maximum of the slope of a solution, the author gives the precise blowup scenario for gCH equation: the slope of the solution becomes unbounded from below in finite time, while the solution remains bounded. The paper is organized as follows. In Sec. 2, the local well-posedness result for the initial value problem for gCH equation is recalled. Sect. 3 is devoted to obtaining the precise blowup scenario for strong solutions to the equation. In Sec. 4 the author presents some criteria ensuring that some classical solutions have a finite life-span, provided their initial data satisfy certain conditions.

35G25Initial value problems for nonlinear higher-order PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B40Asymptotic behavior of solutions of PDE
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