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Global existence for a new periodic integrable equation. (English) Zbl 1033.35121

Summary: We establish the local well-posedness for a new periodic integrable equation \[ \begin{alignedat}{2} &u_t- u_{txx}+ 4uu_x= 3u_x u_{xx}+ uu_{xxx},\quad &&t> 0,\;x\in\mathbb R,\\ & u(0,x)= u_0(x),\quad &&x\in\mathbb R,\\ & u(t, x+1)= u(t,x),\quad &&t\geq 0,\;x\in\mathbb{R}.\end{alignedat} \] We show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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