Some remarks on the inhomogeneous biharmonic NLS equation. (October 2022) - Guzmán, Carlos M.; Pastor, Ademir
Abstract: We consider the inhomogeneous biharmonic nonlinear Schrödinger equation i u t + Δ 2 u + λ | x | − b | u | α u = 0, where λ = ± 1 and α, b > 0 . In the subctritical case, we improve the global well-posedness result obtained in Guzmán and Pastor (2020) for dimensions N = 5, 6, 7 in the Sobolev space H 2 ( R N ) . The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, α = 8 − 2 b N − 4 are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space H ̇ 2 .