Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Abstract

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equationFirst, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian (−∆) s/2 with s ∈ (0, 1] established by [1] to any s > 0. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in H s for the IBNLS equation in both of subcritical case σ < σ c (s) and critical case σ = σ c (s). We also prove that the IBNLS equation is globally well-posed in H s , if the initial data is sufficiently small and 8−2b d≤ σ ≤ σ c (s) with σ < ∞.