We study the Cauchy problem for the higher-order nonlinear 3D Hartree-type equation

$$\begin{aligned} \left\{ \begin{array} [c]{ll} i\partial _{t}u+\frac{1}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\left( \left| x\right| ^{-1}*\left| u\right| ^{2}\right) u,&{}{}\quad t>0, x\in {\mathbb {R}}^{3},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}{}\quad x\in {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$

where \(*\) denotes the convolution in \({\mathbb {R}}^{3}\). Cubic Hartree-type nonlinearity usually behaves critically for large time in three-space-dimensional case. In the present paper, we develop the factorization technique for the case of the higher-order Hartree-type equation (1.1) to show that the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Hartree-type equation has a modified character.

中文翻译：

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高阶非线性Hartree型方程的修正散射

我们研究高阶非线性3D Hartree型方程的柯西问题

$$ \ begin {aligned} \ left \ {\ begin {array} [c] {ll} i \ partial _ {t} u + \ frac {1} {2} \ Delta u- \ frac {1} {4} \ Delta ^ {2} u = \ left（\ left | x \ right | ^ {-1} * \ left | u \ right | ^ {2} \ right）u，＆{} {} \ quad t> 0 ，x \ in {\ mathbb {R}} ^ {3}，\\ u \ left（0，x \ right）= u_ {0} \ left（x \ right），＆{} {} \ quad x \在{\ mathbb {R}} ^ {3}中，\ end {array} \ right。\ end {aligned} $$

其中\（* \）表示\（{\ mathbb {R}} ^ {3} \）中的卷积。三次Hartree型非线性通常在三维空间情况下长时间处于临界状态。在本文中，我们针对高阶Hartree型方程（1.1）的情况开发了因式分解技术，以表明高阶非线性Hartree型方程Cauchy问题解的较大时间渐近性修改后的字符。