Scattering theory below energy for the cubic fourth-order Schrödinger equation

We investigate the global existence and scattering for the cubic fourth-order Schrödinger equation $iu_t+Δ^2u+|u|^2u=0$ in the low regularity space $H^s(\R^n)$ with $s<2$. We provide an alternative approach to obtain a new interaction Morawetz estimate and extend the range of the dimension of the interactive estimate in Pausader \cite{P08} by modifying a tensor product method appeared in \cite{CGT}. We combine interaction Morawetz estimates, energy increments for the I-method to prove the result.

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