Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations
Ebru Toprak, Mengyi Xie
TL;DR
This work establishes small-data scattering for the five-dimensional fourth-order nonlinear Schrödinger equation with quadratic terms, $i\partial_t u+\Delta^2 u+\alpha u^2+\beta\bar{u}^2=0$. It extends the space-time resonance method of Germain–Masmoudi–Shatah to the biharmonic setting, incorporating fractional integration, Coifman–Meyer operators, and flag-singularity analysis to handle bilinear and trilinear interactions. The authors construct a robust $X$-norm framework around the profile $f=e^{it\Delta^2}u$, prove a linear decay $\|e^{it\Delta^2}f\|_{\infty}\lesssim t^{-5/4}$ with refined remainder estimates, and decompose the nonlinear term into resonant and non-resonant parts ($g$ and $h$) to obtain sharp weighted $L^2$ bounds. By a careful bootstrap argument, they close the estimates and deduce global existence and scattering to the free evolution, with the key decay $\|u(t)\|_{L^{\infty}_x}\lesssim t^{-5/4}$. This advances the understanding of high-dispersion, low-homogeneity nonlinear dynamics in higher dimensions and demonstrates the adaptability of space-time resonance techniques to higher-order PDEs.
Abstract
We prove small data scattering for the fourth-order Schrödinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+Δ^2 u+αu^2 + β\bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $α, β\in \mathbb{R}$. We extend the space-time resonance method, originally introduced by Germain, Masmoudi, and Shatah, to the setting involving the bilaplacian. We show that under a smallness condition on the initial data measured in a suitable norm, the solution satisfies $\|u\|_{L^{\infty}_x }\lesssim t^{-\frac{5}{4}} $ and scatters to the solution to the free equation. Although our work builds upon an established method, the fourth-order nature of the equation presents substantial challenges, requiring different techniques to overcome them.