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

Yin, Zhaoyang

J. Math. Anal. Appl. 283, No. 1, 129-139 (2003).

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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