The Cauchy problem for the fourth-order Schrödinger equation in H s

Abstract

We consider the fourth-order Schrödinger equation i∂_tu + Δ²u + μΔu + λ|u|^αu = 0 in $H^s\left({\mathbb{R}}^N\right)$ , with $N\ge 1,\lambda \in \mathbb{C}$ , μ = ±1 or 0, 0 < s < 4, 0 < α, and (N − 2s)α < 8. We establish the local well-posedness result in $H^s({\mathbb{R}}^N)$ by applying Banach’s fixed-point argument in spaces of fractional time and space derivatives. As a by-product, we extend the existing H² local well-posedness results to the whole range of energy subcritical powers and arbitrary $\lambda \in \mathbb{C}$ .