群不变解在基态以下的散射：四阶非线性薛定谔方程

中文摘要 AI 辅助

我们考虑聚焦、$L^2$-超临界且$\dot{H}^2$-次临界的非线性四阶薛定谔方程。Guo [Comm. Partial Differential Equations (2016)] 和 Dinh [Nonlinearity (2021)] 证明了径向对称解在基态以下的散射。本文将散射结果推广到群不变解。在 Komada--Masaki [Nonlinearity (2024)] 中，群不变解在基态以下的散射在某个假设下被证明。为了去除该假设，我们建立了针对一般解的非最优散射结果，其中作用量的阈值小于基态作用量的某个分数。该结果类似于 Pausader--Shao [J. Hyperbolic Differ. Equ. (2010)] 对 $L^2$-临界非线性四阶薛定谔方程的结果。

英文摘要

We consider the focusing, $L^2$-supercritical and $\dot{H}^2$-subcritical nonlinear fourth-order Schrödinger equation. The scattering of radially symmetric solutions below the ground state was proved by Guo [Comm. Partial Differential Equations (2016)] and Dinh [Nonlinearity (2021)]. In this paper, we extend the scattering results to group-invariant solutions. In Komada--Masaki [Nonlinearity (2024)], the scattering of group-invariant solutions below the ground state was proved under a certain hypothesis. To remove the hypothesis, we establish the non-optimal scattering result for general solutions, where the threshold of action is less than certain fraction of the action of the ground state. This result is analogous to that in Pausader--Shao [J. Hyperbolic Differ. Equ. (2010)] for the $L^2$-critical nonlinear fourth-order Schrödinger equation.

发表机构

• Research Institute for Science and Engineering, Waseda University（早稻田大学理工学研究科）

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