Global Solutions for 5D Quadratic Fourth-Order Schr\"{o}dinger Equations

We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity {equation*} i∂_t u+Δ^2 u+α u^2 + β ū^2=0 R^5 {equation*} for α, β ∈ 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∞ₓ t-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.