On the decay property of the cubic fourth-order Schrödinger equation

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on R d \mathbb {R}^d ( 5 ≤ d ≤ 8 5 \leq d \leq 8 ) enjoys the same decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result by Pausader [J. Funct. Anal. 256 (2009), pp. 2473–2517]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.