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 HsRN, with N≥1,λ∈C, μ = ±1 or 0, 0 < s < 4, 0 < α, and (N − 2s)α < 8. We establish the local well-posedness result in Hs(RN) 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 λ∈C.