Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations

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 $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{R}$. 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 $\Vert u{\Vert }_{L_x^{\infty }}\lesssim t^{-\frac{5}{4}}$ 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.

59 pages