2022/11/21 by Van Duong Dinh, Dinh, Van Duong, Sahbi Keraani +1 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2211.11824
openalex publication_date 2022/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a class of biharmonic nonlinear Schrödinger equations with a focusing inhomogeneous power-type nonlinearity i∂t u -Δ2 u+μΔu +|x|-b |u|αu=0, . u|t=0=u0 ∈ H2(ℝd) with d≥ 1, μ≥ 0, 0<b>0, and α<(8-2b)/(d-4) if d≥ 5. We first determine a region in which solutions to the equation exist globally in time. We then show that these global-in-time solutions scatter in H2(ℝd) in three and higher dimensions. In the case of no harmonic perturbation, i.e., μ=0, our result extends the energy scattering proved by Saanouni [Calc. Var. 60 (2021), art. no. 113] and Campos and Guzmán [Calc. Var. 61 (2022), art. no. 156] to three and four dimensions. Our energy scattering is new in the presence of a repulsive harmonic perturbation μ>0. The proofs rely on estimates in Lorentz spaces which are properly suited for handling the weight |x|-b.</b>