Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

2022/10/13 by Luccas Campos, Campos, Luccas, Simão Correia +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2210.07060

openalex publication_date 2022/10/13 · openalex created_date 2022/10/15 · openalex updated_date 2026/07/28

Abstract

We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation, iu_t + Δu +μ|x|^-b|u|^αu=0, u₀∈ H^s(\mathbb R^N) or u₀ ∈ H ^s(\mathbb R^N), with μ=± 1, b > 0, s≥ 0 and 0 < α≤ (4-2b)/(N-2s). By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator.

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