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, et
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}} $ et
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.

Cet article explore les excursions dans le temps et leurs implications.

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