## 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.)

### Keywords:

local well-posedness; blow up; global existence
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### References:

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