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On the Cauchy problem for a nonlinearly dispersive wave equation. (English) Zbl 1021.35100

Summary: We establish the local well-posedness for a new nonlinearly dispersive wave equation \[ u_t-u_{txx}+2\omega u_x+3uu_x=\gamma(2u_xu_{xx}+uu_{xxx}),\quad t>0,\;x\in\mathbb{R}, \] and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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