Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrödinger equation in two space dimensions i∂tu + 1 4Δ2u = λ|u|2u,t > 0,x ∈ ℝ2,u(0,x) = u0(x),x ∈ ℝ2, where λ > 0. We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition ∫ℝu0(x)dx≠0. We continue to develop the factorization techniques. The crucial points of our approach presented here are the L2 — estimates of the pseudodifferential operators and the application of the Kato–Ponce commutator estimates.

中文翻译：

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二维四阶非线性薛定谔方程的渐近

我们的目标是研究二维空间维度上四阶非线性薛定谔方程解的大时间渐近性 一世∂吨你 + 1 4Δ2你 = λ|你|2你,吨 > 0,X ∈ ℝ2,你(0,X) = 你0(X),X ∈ ℝ2, 在哪里λ > 0.我们证明非线性具有耗散特性，因此如果我们假设非零总质量条件，则与相应的线性情况相比，解获得更快的时间衰减率∫ℝ你0(X)dX≠0.我们继续开发分解技术。我们这里介绍的方法的关键点是大号2— 伪微分算子的估计和 Kato-Ponce 换向器估计的应用。