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.


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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