We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity \begin{ equation*} i\partial_t u+\Delta^2 u+\alpha u^2 + \beta \bar{ u}^2=0\qquad\text{ in }\mathbb{ R}^5 \end{ equation*} for $\alpha, \beta \in \mathbb{ R}$. 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.
Categories: math.AP